Solved Examples on Derivative of ex
Some examples related to Derivative of ex are,
Example 1: Find the derivative of e2x.
Solution:
Using Chain Rule
y = e2x
y’ = d(e2x)/dx
y’ = [d(e2x)/dx ].[ d(2x)/dx]
y’ = e2x. 2
y’ = 2(e2x)
Example 2:Find the derivative of ex/x2.
Solution:
Using Quotient Rule
[(d[u/v]/dx) = [u’v – uv’]/v2]
y = ex/x2
y’ = d(ex/x2)/dx
y’ = ( d(ex)/dx . x2 – ex. d(x2)/dx )/ (x2)2
y’ = ( ex. x2 – ex. 2x )/ x4
y’= ( x .ex( x – 2))/x4
y’ = ( ex. (x-2) )/ x3
Example 3: Find the derivative of the function (ex)x
Solution:
y= (ex)x
Taking log both side, we get,
ln y = ln (ex)x
ln y = x.(ln ex )
ln y = x2
Differentiation both side, we get,
1/y . y’ = 2x
y’ = y. (2x)
y’ = (ex)x. (2x)
Example 4: Evaluate the derivative of e2x+ x.sinx
Solution:
y = e2x + x.sinx
y’ = d(e2x)/dx + d(x.sinx)/dx
To solve this we need to apply Chain rule for e2x and Product rule for x.sinx
y’ = (d(e2x)/dx . d(2x)/dx ) + (dx/dy. sinx + x . d(sinx)/dy)
y’ = e2x.2 + (1.sinx + x.cosx)
y’ = 2.e2x + sinx + xcosx
Differentiation of e to the Power x
Derivative of ex is ex. Derivative of ex 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 ex is ex. Understanding the derivative of ex 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 ex, what is derivative, and its some basic rules.