If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

To log in and use all the features of Khan Academy, please enable JavaScript in your browser.

Main content

Class 12 math (India)

Course: Class 12 math (India) > Unit 5

Lesson 14: Exponential functions differentiation

© 2023 Khan AcademyTerms of use Privacy Policy Cookie Notice

Differentiating exponential functions review

Google Classroom

Review your exponential function differentiation skills and use them to solve problems.

How do I differentiate exponential functions?

First, you should know the derivatives for the basic exponential functions:

\frac{d}{d x} (e^{x}) = e^{x}

\frac{d}{d x} (a^{x}) = \ln (a) \cdot a^{x}

Notice that

e^{x}

is a specific case of the general form

a^{x}

where

a = e

. Since

\ln (e) = 1

we obtain the same result.

You can actually use the derivative of

e^{x}

(along with the chain rule) to obtain the general derivative of

a^{x}

.

Want to learn more about differentiating exponential functions? Check out this video.

Practice set 1: exponent is $x$ ‍

Problem 1.1

f (x) = - 4 e^{x}

f^{'} (x) = ?

Want to try more problems like this? Check out this exercise.

Practice set 2: exponent is a polynomial

Problem 2.1

y = e^{(3 x^{2} - 4)}

\frac{d y}{d x} = ?

Want to try more problems like this? Check out this exercise.

Want to join the conversation?

Sort by:

Krystina Huculak
Posted 6 years ago. Direct link to Krystina Huculak's post “I need help with differen...”
I need help with differentiating the equation y= xe^(5x) because I need to use the First Derivative Test in order to find the local extrema, however, I'm having trouble understanding how to do the differentiation of the equation.
Button navigates to signup pageButton navigates to signup page
(2 votes)
Answer
- Mark Geary
  Posted 6 years ago. Direct link to Mark Geary's post “Let f(x) = x, and g(x) = ...”
  Let f(x) = x, and g(x) = e⁵ˣ. Use the Product Rule: d/dx f(x)g(x) = f'(x)g(x) + f(x)g'(x). Next let u(x) = eˣ and v(x) = 5x, then use the Chain Rule: u'[v(x)]v'(x).
  Button navigates to signup page
  (3 votes)
anaratf
Posted 3 years ago. Direct link to anaratf's post “How can we differentiate ...”
How can we differentiate e^x^x ? Or similar questions with double powers of exponential functions?
Button navigates to signup pageButton navigates to signup page
(2 votes)
Answer
- famousguy786
  Posted 3 years ago. Direct link to famousguy786's post “You will have to use the ...”
  You will have to use the chain rule. First differentiate the whole function with respect to e^x, then multiply it with the differentiation of e^x with respect to x. You'll solve it. Basically every composite function can be differentiated using the chain rule so that should be the first approach to take.
  Button navigates to signup page
  (1 vote)
TanaOno
Posted a year ago. Direct link to TanaOno's post “Can someone help me with ...”
Can someone help me with this question?

If f(x)=e^(2/x), then f'(x)=
Button navigates to signup pageButton navigates to signup page
(1 vote)
Answer
Pheebs11
Posted 6 years ago. Direct link to Pheebs11's post “is there an easy way to r...”
is there an easy way to remember which kind of question goes to which type of answer?
Button navigates to signup pageButton navigates to signup page
(0 votes)
Answer
londekamemela997
Posted 7 years ago. Direct link to londekamemela997's post “differentiate the followi...”
differentiate the following 7^3x+2
Button navigates to signup pageButton navigates to signup page
(0 votes)
Answer
- kubleeka
  Posted 7 years ago. Direct link to kubleeka's post “d/dx(7^(3x)+2) d/dx(e^ln(...”
  d/dx(7^(3x)+2)
  d/dx(e^ln(7^(3x)) +2) (e^x and ln(x) are inverse functions, so we can apply them together like this)
  d/dx(e^(3xln(7)) +2) (properties of logarithms)
  d/dx(e^(3xln(7)))+d/dx(2) (linearity of derivatives)
  e^(3xln(7))•3ln(7)+0 (derivative of e^x and Chain Rule, derivative of a constant)
  e^(ln(7^(3x)))•3ln(7) (property of logarithms)
  7^(3x)•3ln(7) (e^x and ln(x) are inverse functions)
  Button navigates to signup page
  (1 vote)
Indreta08
Posted 6 years ago. Direct link to Indreta08's post “Can someone help me to di...”
Can someone help me to differentiate this : f (x)= xe^(-x^2/2)
Button navigates to signup pageButton navigates to signup page
(0 votes)
Answer
- Mark Geary
  Posted 6 years ago. Direct link to Mark Geary's post “Point of clarification: i...”
  Point of clarification: in the exponent of e, is it (-x^(2/2)), or is it ((-x^2)/2)?
  Assuming it is the latter, because the former simplifies to (-x),
  d/dx xe^((-x^2)/2) = xe^((-x^2)/2) * -x
  It is a chain rule problem, with d/dx (ae^n)=(ae^n)*(dn/dx)
  Button navigates to signup page
  (1 vote)