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How To Find Derivative Of A Function? [Simple Ways]

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How To Find Derivative Of A Function?

Maths comes in use in our daily life activities in various purposes such as to calculate discounts, banking, statistics, to find the dimensions of an object (such as to find the length, width, altitude, and so on), and many more. Most of the students often feel that mathematics is the most challenging and difficult subject among all. If you are a student, then I hope that you might have faced several issues at the time of your maths exams in solving derivation and integration problems. To find derivative of a function can be useful in many ways. For example: to calculate velocities, to compute forces etc.

The world of maths is so vast that sustaining it all in a single book is not possible. Subdivision of this broad subject includes several fields like Algebra, Calculus, Geometry and many more. Beyond all types of mathematical textbooks, it finds a wide application in practical life as well. Out of this, calculus is a branch that stays in the lives of students from high school to college and probably in your jobs too.

So, here is a glimpse of the derivatives and a take on the same.

In today’s article, we are going to know about a few methods through which we can find derivative of a function.

What is a Derivative?

A derivative is defined as the rate of one quantity with respect to the other. There are two variables- one in the denominator called the dependent variable and the other with respect to which change occurs called the independent variable.

Derivative of x in respect to time is∶ dx/dt=□((change in position)/(change in time))

Where x is denoting position and t is denoting time.

The derivative is used for measuring the steepness of the graph of a function at a particular point, hence also called the derivative is a slope. However, the slope of a secant line approaches the derivative when the interval between the points shrinks down to zero.

The derivative of a derivative also exists. For example, the derivative of a position with respect to time is called velocity. Further, the derivative of velocity with respect to time is called acceleration.

How to calculate Derivatives?

To find derivative of a function, you basically need to calculate the change in both the quantities and then divide them to get the derivative.

To find the derivative, we use the slope formula

Slope = □((change in y)/(change in x)=Δy/Δx)

However, there are plenty of formulas that reduce the burden of finding the slope by this congested method.

Basics one should know before starting to deal with Derivatives

Now that we are clear about the derivatives let’s get deeper into the basics revolving around it. The basic has been described above. However, to simplify the use of derivatives, several formulas have been formulated for the slope method gets cumbersome many a time.

Derivative Of A Function

The derivative of a function helps to find the sensitivity to the change in the function value(output) with respect to the change in its argument(input). For example, the derivative of the state of a traveling object with respect to time is the velocity, which describes how fastly the position of the object refers as time changes.

The method of finding a derivative is called differentiation, and its reverse process is called antidifferentiation. The derivative calculates the steepness of the graph of the function, at some particular point on the graph. If we zoom that end, then we may notice a straight line called the slope of the line.

The derivative is simply defined as the instantaneous rate of change, slope, at a particular point of a function. The derivative is denoted by (dy/dx), which stands as the derivative of y(output) with respect to x(input).

Let us have a peep at some of the methods through which you can find derivative of a function.

The derivative of a number

The derivative of a number or a constant is always zero as it does not change in the function. Some of the examples are.

  • (4)’=0(a derivative of 4 is 0)
  • (pi)’=0(pi=3.14, a constant value, so the derivative will be zero)

Algebraic function

To find derivative of a function which is algebraic in nature, the simple rule says, get the power in front as a constant and reduce the power number by 1.
d/dx 〖(x〗^n)=nx^(n-1), here n denotes any real number.

For example: d/dx(x^4)=〖4x〗^3

A number multiplied with a variable without any exponent.
The derivative of any form which does not have any exponent is the number itself.

Examples

  • (4x)’=4( the derivative of 4x is 4 where x is variable)
  • (x)’=1(the derivative of x is 1)

A number multiplies with a variable with an exponent

We often face difficulty in finding the derivative of large powers. These are the few steps to be followed to get the derivative of a function with exponents.

Example: (4x^3)=12x^2

Steps:

  1. Multiply the number with the value available at power. (4*3=12)
  2. Subtract one from the exponent. (3-1=2)

Simply the function

Functions are of many types. To find the derivative of any function, we have to convert the complex function into a more straightforward form for fruitful results. Let us see how we can resolve it.

Consider a function as (4x+5x)/3+2x+5

In order to find the derivative of the above function, we have to eliminate the size of the function as follows:

  • 4x+5x=9x(adding numerators)
  • 9x/3=3x(dividing numerator and denominator)

The function will be simplified as 3x+2x+5

Later, the result is 5x+5, and its derivative is 5(the derivative of 5x is 5, and the derivative of constant is 0)

Trigonometric Functions

Let’s now look to find derivative of a function involving trigonometry. We use trigonometric functions such as sine, cosine, tan, cot, secant, and cosecant in mathematics. In a similar fashion, we can also find the derivatives of trigonometric functions. Some of them are listed below. To deal with trigonometric functions, you basically need to cram a few formulae whose origin obviously lies in the slope formula.

  • d/dx (sin⁡x )=cos⁡x
  • d/dx(cos⁡〖x)=-sin⁡x 〗
  • d/dx(tan⁡〖x)=〖sec〗^2 x〗
  • d/dx(sec⁡〖x)=sec⁡〖x.tan⁡x 〗 〗
trigonometric functions
trigonometric functions

Above are the derivatives of the basic trigonometric functions used. The one for cosine, cotangent can be used by converting them to sine and cosine.

Rules to Find Derivative Of A Function

You can also find derivative of a function through some regulations in an effortless manner just by following few steps. Let us have a look at it.

Power Rule

The power rule states that the derivative of (x^n)’=(n*x^n-1). This rule is applicable for polynomials as well.

  • (f+g)’=(f)’+(g)’
  • (f-g)’=(f)’-(g)’
Power rule
Power rule

Example:

(45y^2+25y+3)’=(90y+25)(45*2=90 and the derivative of 25y is 25 where y is variable)

Product Rule

This product rule is applicable for all functions and can be given by the formula.

d/dx(f(x)g(x))= f'(x)g(x)+f(x)g'(x)

  • where f(x) is the first function
  • g(x) is the second function
  • f'(x) is the derivative of the first function and
  • g'(x) is the derivative of the second function.
product rule
product rule

Here the two variables are dealt with separately in parts where one is assumed t be constant, and the other is differentiated and so on the rule continues till the time all the parameters are differentiated.

d/dx (FG)=F d/dx (G)+G d/dx(F)

Where F and G are functions.

For example, d/dx (sin⁡x*x^3 )=sin⁡〖x*3x^2+x^3*cos⁡x 〗

Example:

The derivative of ((3x^2)(9x+5))’

  • Let f(x)=3x^2, g(x)=9x+5
  • f'(x)=6x and g'(x)=9
  • Arranging in the product rule formula then we get
    (6x)(9x+5)+(9)(3x^2)
  • on further simplification:
    54x^2+30x+9(6x)

Quotient Rule

This rule is applicable if the function is in the numerator and denominator(f/g) form.

(f(x)/g(x))’=((g(x)*f'(x)-f(x)*g'(x))/g(x)^2

quotient rule
quotient rule

Let us have a peep at an example:

The derivative of ((4x^3)/(8x^2))’

  • Let us consider f(x)=4x^3, g(x)=8x^2, then
  • f'(x)=12x^2, g'(x)=16x
  • So the final derivative form is
    (((8x^2)*(12x^2))-((16x)(4x^3)))/((8x^2))^2

Chain Rule

Now here comes the fun part. Suitable for composite functions, this rule is very helpful in the long run.

If F and G are two functions of x, the composite function F(G), with G being the Parameter of F is calculated for a value of x by first evaluating G and then, assess the function F at this value of G.

d/dx F(G)=d/dx F(G).d/dx G

Here, F(G) denotes that G is the parameter of Function F.

Chain rule
Chain rule

For example, dy/dx sin⁡(3x^2 )=cos⁡〖3x^2*6x〗.

This rule is appropriate to find derivative of a function whose form will be as g(f(x)). Let us consider an example.

If you wish to see the derivative of a function such as cos(x^2+5), then you can opt for the chain rule. Following the below-mentioned steps:

Example:

  • The derivative of (sin(4x))’
  • At first, we have to find the derivative of sine which is cos
    (sin4x)’=cos4x( the derivative of sin4x is cos4x)
  • Next, we have to find the derivative of the inner parenthesis of the function
    (4x)’=4(the derivative of 4x is 4)
  • And have to multiply the result obtained in the above two steps. Hence, the derivative is 4 cos(4x)

Sum Rule

Take out the constant from the main function.

d/dx (aF+bG)=a d/dx F+b d/dx G

Where a and b are constants and F and G are functions of variable x.

Sum Rule
Sum Rule

For example, d/dx (3x^2+4x^4 )=3*2x+4*4x^3
The same rule applies to the subtraction of derivatives.

Division rule

Here the square of the denominator is the denominator of the derivative, and the numerator follows the same procedure as the product rule except for the fact that the other function is the denominator followed by the subtraction of the same.
d/dx (F/G)=(G d/dx (F)-F d/dx(G))/G^2
Here, F and G are two functions of x.

For example, d/dx (x^2/sin⁡x )=sin⁡〖x*2x-x^(2 ) cos⁡x 〗/(〖sin〗^2 x)

Explicit Differentiation

This approach is handy for explicit functions where the derivative computation goes through differentiating the same function in parts with different independent variables and then dividing.

dy/dx F(G)=dF/dG.dG/dx

Here, F(G) denotes that G is the parameter of function F.

Conclusion

These basics form the basis of differentiation. If one has mastered this, he has aced this topic. Remember, differentiation is one topic that will be in use for a lifetime. Everything, primarily velocity, acceleration, change of the rate of time needs this part of maths. And that is why you find calculus Jest of Mathematics.

I hope that you all have got an impression on how to find derivative of a function. Several derivative calculators are available online through which you can find the derivative of any function by just entering the function in the search bar. You can also go up with the above methods and rules to find derivative of a function.

So, what are you waiting for? Go and sit down to try some fun questions on this. Give it a hard hit and even then if things don’t work out, come back for the revision but don’t give up. Remember, Practice makes a man perfect.

How To Find Derivative Of A Graph | Easiest Method

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How To Find Derivative Of A Graph

If you are a science stream guy, then derivative has become a part of your daily routine most probably. As you start studying mathematics, physics, and chemistry in high school, immediately you get introduced to this significant agent of mathematics, then if you plan to continue your studies in technology area possibly in engineering. The derivatives especially to find derivative of a graph is something which you will explore in this article and it will be mixed up with your soul permanently.

Whether it be physics or chemistry, applications of derivatives and to find derivative of a graph is everywhere. Most of the times, derivative are closely related to the graphs. It is possible to calculate the derivative of a function from its plot. The result of a derivative calculated from a graph may not have the same accuracy as calculated from an original equation. But you can surely have an idea of the rate of change of the function from the derivative of the graph.

What is a Derivative?

A derivative an implement of calculus which is a part of mathematics. It is a regular requisition for estimating several aspects of science. A derivative generally represents the rate of change of a given function. When you change the input of a particular function, the output also changes gradually. This scenario is represented by the derivative.

tangent of a graph
tangent of a graph

A function consists of an input and an output segment. The derivative helps in establishing the relation of improvement between them. It shows the dependency of the output of the function on the input section. It can be shown manually. There is a lot of software also available to determine the derivative of a function.

Application of Derivatives

If you are from a science background, you must know the solemnity of derivative in the field of science. It may be the formula of velocity or the reaction rate of a chemical experiment. One can use derivatives to show the result mathematically. There is more utilization of derivatives in the domain of science and technology.

The applications are:

  • In physics, the derivative is used to estimate the rate of change. For example, the change of distance with respect to time or the rate of change of speed with the respect of time is known as velocity and acceleration, respectively. This particular can also be analyzed if one is able to find derivative of a graph.

    `
    Physics: Applications of derivatives
  • In order to obtain critical points of a function, it’s crucial to find derivative of a graph.
  • From the derivative of a graph, one can also obtain the maximum and minimum value of a function.
  • We may use the derivative of a graph in the linear approximations in specific points of a function.
  • A derivative is also in use with several theorems and formulae like mean value theorem, L’ Hospital rule, and Rolle’s theorem.

Steps to Find Derivative Of A Graph

To find derivative of a graph, you need to follow these steps:

Selection of the appropriate point for derivative

At first, you have to choose a suitable point on your curve for estimation of the derivative. The selected point will show the slope of the graph on itself. As derivative indicates the gradient of the plotting, so it will change on each and every point of the curve.

Draw a Tangent on the Chosen point

After picking up the point on the graph, you have to go for the tangent representation on the point. The tangent is a straight line, and it must be touched on a single point on the curve, it will evince the accuracy of the curve. However, it is a tiresome and tough action to perform the gradient line determination.

Tangents at multiple points
Tangents at multiple points of a curve

The more the line touches on a single point, the more is the accuracy of the tangent. You may extend the straight line of the slope as long as the curve allows you to do so, but you should note the horizontal and vertical coordinate values simultaneously. More than half of your work is complete as soon as you finish drawing the tangent correctly.

Determination of Derivative from the Tangent

After illustrating the gradient line accurately, you have to put on record the X and Y axis coordinate values of the gradient line in order to find derivative of a graph. The values will indicate the position of the start and end point of the tangent line. Right after acquiring the coordinate values, you can estimate the derivative of the plot on that particular point. To compute it follow the following steps:

A derivative of the graph on a particular point = (The difference between Y-axis coordinate values of tangent) / (The difference between X-axis coordinate values of tangent)

As the coordinate values of the slope are the governing framework of derivate calculation, so it is easy to interpret that the derivative will change from point to point on the graph. Thus a derivative can indicate the rate of change of a function.

Roadmap to Find Derivative Of A Graph Accurately

To find derivative of a graph ideally, you have to draw the tangent line precisely first.

  • Use a sharp headed pencil to draw the gradient line on the curve.
  • You should use a straight and sharp-edged ruler to draw the tangent.
  • The ruler must be in alignment on a single point on the curve. Only thus, you can get the gradient line in orientation on a specific point.
  • Mark the coordinates of the tangent line perfectly.

Following the above steps, you can find derivative of a graph on the selected point.

What is a Partial Derivative? Scope and Application

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What is a Partial Derivative

Differentiation is a common process in mathematics and physics. Differentiation is the process of finding the derivative of a function. This derivative is to compute the rate of change of something, and hence, Partial Derivative has numerous applications in the field of science and mathematics. Partial derivative can be handy to solve real-world problems.

Using derivatives, you can identify the rate of change of volume, area, speed, etc. Derivative usually refers to a function with a single driven variable, i.e., The function has a single dependent and single independent variable.

Learning and understanding derivatives can save you from multiple complex Mathematics problems. The more you understand, the better you become at it. Here, we will teach you some tips and tricks for elementary studying and competence.

So, the derivative can either be written as d(dependent variable) / d(independent variable). If y is dependent and x is independent, it can be written as:

dy/dx.

If x is dependent and Y is independent, it can be written as:

dx/dy.

A derivative of a function is usually computed using ‘The First Principle’ as shown below.

f’(a) = lim (f(a+h) – f(a)) /h

where the limit is h0.

Now, if you’re dealing with more complex functions or equations you might come across a term caked partial Derivative. What is this partial Derivative? Where to use it? Let’s find out.
Partial Derivative is the result of the differentiation of a function with several variables with respect to a single variable. To be precise, a function with many independent variables.

This makes the process of differentiation a bit more difficult. Both partial differentiation and differentiation comes under the topic calculus, which is the heart of mathematical physics.

Notation of Partial Derivative

The operator of the partial Derivative is denoted by ∂ and can be pronounced as ‘dou’. In some regions, it is also pronounced as ‘dough,’ but the former one is in use in most of the cases.

Now, a partial Derivative of a function containing variables a, b, c, d, and so on up to z concerning x is as follows:

∂a/∂x + ∂b/∂x + ….. + 1 +.. + ∂z/∂x

General representation is ∂f/∂x, where f is a function with many independent variables. The general notation found in most of the textbooks for ∂f/∂x is p and for ∂f/∂y is q. These notations need not be the same in every textbook.

Concept Of Derivatives

For becoming a pro at roots of calculus, you must have the basic knowledge and know
the concept of derivatives. It is the essence of Calculus.

    • It is defined as the ratio between the variation of two variables, that is, x and y.

      Slope = Change in y/Change in x

Partial Derivative Definition

Partial Derivatives are similar to limits. The First Principle is to apply for functions with many independent variables, and the partial Derivative is easier to obtain. Partial Derivative is similar to the actual derivative in many cases.

We can check differentiability at a given point using partial derivatives just as in the case of derivatives. If all the derivatives of a function ‘f’ at a point a exist in the neighborhood of the given point ‘a’ and are continuous at the same point, then the function can be said differentiable function.
However, continuity is still uncertain at a point by observing the partial derivatives. A function need not be continuous at a point ‘a’ even if all the partial derivatives of the function exists at the same point.

partial derivatives representation
partial derivatives representation

Therefore, you need to find out the derivative to determine continuity. However, we know that a function should be continuous to be drivable. Hence we can justify the continuity of a function in this way.
Double differentiation is also applicable for functions with multiple independent variables. This means that just as in the case of derivatives, there can be double partial derivatives. To depict double partial derivatives we use (∂^2) f/(∂x) ^2, and which sounds “dou Square f by dou x Square.”

See Also: How to Ping a Cell Phone | Step by Step Guide

We can differentiate a partial derivative of x, i.e., ∂f/∂x  with respect y giving (∂^2) f/∂x∂y. which pronounces as “Dou square f by dou x dou y.” In many textbooks, representation is with the letter ‘s’ which need not be the same in every textbook.

Solving Partial Derivatives

Follow the same conventional way of normal derivatives to solve partial derivatives.  If you need to solve the function f= 5x + 6y where x and Y are functions of another variable ‘t,’ you can solve the function as follows:

∂f/∂x = 5+6×∂y/∂x
∂f/∂y = 5×∂x/∂y + 6

If you want to find the complete derivative or the total derivative of the function, you can solve the values and substitute in the below formula.

f’(x) = ∂f/∂x × dx/dx + ∂f/∂y × dy/dx

This formula gives you the final derivative of the function f with respect to x. If you want to find out the total derivative of the function f(x, y) with respect to y, you can multiply ∂f/∂x with dy/dx and ∂f/∂y with dy/dy which is one.

Applications of Partial Derivatives

Partial derivatives are an important part of every optimization problem with two or more variables. Since partial Derivatives operate on multiple variables, they extensively operate in the Firefox of mathematics and science.

Many Engineering applications employ the use of Partial derivatives. Using Partial derivatives is common in the field of economics when operating with multiple variables.
The image resizing software also employs the use of partial derivatives to estimate the energy of pixels.

Maxima and Minima using Partial Derivative
Maxima and Minima using Partial Derivative

Thermodynamics also has many equations with multiple variables which makes the partial derivatives key components in solving thermodynamic equations.

Since many equations in thermodynamics employ the use of partial derivatives, applications of the partial derivatives extend to the field of Astrophysics. With Partial derivatives, we can also find the relative maximums and minimums in mathematics. Moreover, we can compute absolute Maxima and absolute Minima using partial derivatives as well.

What is a derivative? Everything About it in One Post!

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What is a derivative

The biggest nightmare for half of the student population is Mathematics. The hefty theorems, properties, tricks, formulas, etc. always ended up giving them a headache. The never-ending terminologies, assumptions, exceptions, etc. perpetually scared the hell out of them. The infinite bucket list of Mathematics consisted of a new adventure for them, namely Derivatives. So let us explore What is a derivative?

Derivatives are uncomplicated and very easy to understand if evaluated with proper focus and understanding. Comprehending the art of Mathematics can be precarious, but once mastered; it can turn out to be the best part.

Still you must be confused that What is a derivative? Learning and understanding derivatives can save you from multiple complex Mathematics problems. The more you understand, the better you become at it. Here, we will teach you some tips and tricks for elementary studying and competence. Practicing is the foremost key to becoming successful.

The derivative is the heart of Calculus. It has contributed to a significant part of Calculus Mathematics and one of the best fundamental tools of Calculus. Also it shows the rate of change of some function at a given point. Derivatives are most commonly found using differentiation problems. It consists of many rules and formulas which have to be followed thoroughly for having a better grip at the concept of derivatives. Follow the article to find out and understand better that What is a derivative?

Concept Of Derivatives

For becoming a pro at roots of calculus, you must have the basic knowledge and know
the concept of derivatives. It is the essence of Calculus. Here is a mathematical definition to What is a derivative?

  • It is defined as the ratio between the variation of two variables, that is, x and y.
    Slope = (Change in y/Change in x)
  • Shows the rate of change of the amount by which the value of a function is changing at a specific point.
  • Is the slope of the tangent line to the graph of the function of at a particular point.
  • Generally represented by “dy over dx” which means the change in y; is divided by the change in x.
  • The derivative can be precised as: dy/dx, where d cannot be cancelled.

What is a Derivative and How to find them?

The derivative of a function y = f(x) can be found using the slope formula:

Slope = (Change in Y = Δy/Change in x    Δx)

From figure, we can observe that:

slope of a line formula
slope of a line formula

Y changes from: f(x)   to f(x+ Δx)

X changes from: x    to x+ Δx

Now follow further mentioned steps:

  • Now fill the above mentioned slope formula with the changed values of x and y.

Slope = Change in Y/Change in x

Δy/ Δx=

(f(x + Δx) – f(x)/x + Δx – x) =     f(x + Δx) – f(x)/Δx

  • Simplify the given functions to the best possible outcome.
  • Then finally make Δx shrink towards zero.

Formulas Of Derivatives

The derivative of functions has different formulas. There are linear functions, exponential functions, power functions, trigonometric functions, logarithmic functions, and hyperbolic functions.  

Linear Functions

So What is a derivative of a Linear function, but first of all What is a Linear function? It tells that functions of the form ax + b with no higher or quadratic terms are constant.

 

                                                  d/dx(x) =1

Power Functions

They are different than the linear functions because they consist of an exponent value.

 

 

derivative_of_power_function
derivative_of_power_function
derivative_of_power_function
derivative_of_power_function

Exponential Functions

It is in the form of ab^f(x) where a and b are constants and f(x) is a function of x. So What is a derivative of a Exponential function?

derivatives of exponential functions
derivatives of exponential functions

Hyperbolic Functions

 

Hyperbolic Functions
Hyperbolic Functions

Logarithmic Functions

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Trigonometric Functions Derivatives

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Similarly, both trigonometric and hyperbolic functions also have inverse trigonometric and inverse hyperbolic derivative functions.

Properties Of Derivatives

Like other calculus concepts, derivatives also follow some unique properties.

The derivatives can be broken into smaller parts for easy calculation and simplification of equations. 

derivative example
derivative example

Practice More

The more you practice, the better you understand. Always keep researching for new tricks and methods to solve derivatives. In this way, you will learn about new crucial points that will help you in understanding derivatives well. Follow all the steps carefully and patiently. Try to solve the maximum number of questions of different types and styles. After some time, it will be a left-hand play for you to understand derivatives. Now you must have got a clear idea of What is a derivative?

Conclusion

Derivatives can become your favorite part of studying Calculus if grasped with proper focus. It is one of the most enthusiastic topics any mathematician should understand. Put all your efforts into it, and it will give you the best knowledge, tips, and tricks.