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)
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.
- (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.
- Multiply the number with the value available at power. (4*3=12)
- 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)
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 (sinx )=cosx
- d/dx(cos〖x)=-sinx 〗
- d/dx(tan〖x)=〖sec〗^2 x〗
- d/dx(sec〖x)=sec〖x.tanx 〗 〗
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.
The power rule states that the derivative of (x^n)’=(n*x^n-1). This rule is applicable for polynomials as well.
(45y^2+25y+3)’=(90y+25)(45*2=90 and the derivative of 25y is 25 where y is variable)
This product rule is applicable for all functions and can be given by the formula.
- 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.
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 (sinx*x^3 )=sin〖x*3x^2+x^3*cosx 〗
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
- on further simplification:
This rule is applicable if the function is in the numerator and denominator(f/g) form.
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
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.
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:
- 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)
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.
For example, d/dx (3x^2+4x^4 )=3*2x+4*4x^3
The same rule applies to the subtraction of derivatives.
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/sinx )=sin〖x*2x-x^(2 ) cosx 〗/(〖sin〗^2 x)
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.
Here, F(G) denotes that G is the parameter of function F.
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.