Differentiation of Exponential Functions

Differentiation is very important part in mathematics which is the act of finding the derivative of any function and the process of finding the derivative of a function is differentiation.

The differentiation notation is `dy/dx` which means y is any function which is differentiated with respect  to x. Differentiation can be used to find the derivative of various kinds of function such as trigonometric function, logarithmic function and exponential function, inverse trigonometric function, implicit functions, composite functions etc.

Here, in this tutorial we are going to discuss about differentiation of exponential function. Exponential function is a function which involves two numbers, the base and the exponent such as ax where a is the base and x is the exponent.

A function f is said to have a derivative at any point x iff it is defined in some neighbourhood of the point x and `lim_(deltax->o)` `(f(x + deltax) - f(x))/(deltax)` exits ( finitely).

The value of this limit is called the derivative of f at any point x and is denoted by f ' (x) i.e.

f ' (x) =`lim_(deltax->o)`  `(f(x+ deltax)-f(x))/(deltax)`

This tutorial will help student in understanding the concept of differentiation of exponential function if they have any problem in this topic, and the solved problems and practice problem given here will held students to get perfect in derivative of exponential functions.

Formula For Differentiation of Exponential Function:

This formulas will help in finding the differentiation of exponential function.

`d/(dx)` (`e^x)` = `e^x`

`d/(dx)` (`e^(ax))` =a`e^(ax)`

`d/(dx)` (`e^u)` =  `e^u` `(du)/(dx)`

`d/dx` `(b^u)` = `b^u` log b `(du)/(dx)`

`d/dx` `(a^x)` = `a^x` log a


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Problems On Differentiation of Exponential Function:


Problem 1: Differentiate the function y = `e^(3x)` with respect to x.


Given y = `e^(3x)`

Differentiating the above with respect to x we get,

`(dy)/dx` = `d/dx` ( `e^(3x)` )

         = `e^(3x)` `d/dx` (3x)

        = `e^(3x)` * 3

         = 3 `e^(3x)`

Problem2: Differentiate y = `e^x` log x with respect to x.



   y = `e^x` log x

Differentiating the above function with respect to x,

`(dy)/dx` = `d/dx` ( `e^x` log x)

           = `e^x` `d/dx` ( log x) + logx `d/dx` (`e^x` )

           = `e^x` `1/x` + log x `e^x`

           = `e^x` ( `1/x` + log x )


Practice Problems


Problem 1: Differentiate `e^x` cot x with respect to x. ( Answer: `(dy)/dx` = `e^x` ( cot x  - `cosec^2 x` ))

Problem 2: Differentiate `1/3` `e^x` - 2 log x with respect to x ( AnswerL `(dy)/dx` = `1/3 e^x` - `2/x` )