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.

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