Solved Examples on Derivative of ex

Some examples related to Derivative of e^x are,

Example 1: Find the derivative of e^2x.

Solution:

Using Chain Rule

y = e^2x

y’ = d(e^2x)/dx

y’ = [d(e^2x)/dx ].[ d(2x)/dx]

y’ = e^2x. 2

y’ = 2(e^2x)

Example 2:Find the derivative of e^x/x².

Solution:

Using Quotient Rule

[(d[u/v]/dx) = [u’v – uv’]/v²]

y = e^x/x²

y’ = d(e^x/x²)/dx

y’ = ( d(e^x)/dx . x² – e^x. d(x²)/dx )/ (x²)²

y’ = ( e^x. x² – e^x. 2x )/ x⁴

y’= ( x .e^x( x – 2))/x⁴

y’ = ( e^x. (x-2) )/ x³

Example 3: Find the derivative of the function (e^x)^x

Solution:

y= (e^x)^x

Taking log both side, we get,

ln y = ln (e^x)^x

ln y = x.(ln e^x )

ln y = x²

Differentiation both side, we get,

1/y . y’ = 2x

y’ = y. (2x)

y’ = (e^x)^x. (2x)

Example 4: Evaluate the derivative of e^2x+ x.sinx

Solution:

y = e^2x + x.sinx

y’ = d(e^2x)/dx + d(x.sinx)/dx

To solve this we need to apply Chain rule for e^2x and Product rule for x.sinx

y’ = (d(e^2x)/dx . d(2x)/dx ) + (dx/dy. sinx + x . d(sinx)/dy)

y’ = e^2x.2 + (1.sinx + x.cosx)

y’ = 2.e^2x + sinx + xcosx

Derivative of e^x is e^x. Derivative of e^x means finding the change in the exponential function with respect to the independent variable. The process of finding the derivative is known as differentiation. The derivative of e^x is e^x. Understanding the derivative of e^x is an important concept in calculus as it offers insights into the dynamic nature of exponential growth.

In this article, we will talk about derivatives of e^x, what is derivative, and its some basic rules.