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## Class 12 math (India)

### Course: Class 12 math (India)>Unit 5

Lesson 14: Exponential functions differentiation

# Differentiating exponential functions review

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:
$\frac{d}{dx}\left({e}^{x}\right)={e}^{x}$
$\frac{d}{dx}\left({a}^{x}\right)=\mathrm{ln}\left(a\right)\cdot {a}^{x}$
Notice that ${e}^{x}$ is a specific case of the general form ${a}^{x}$ where $a=e$. Since $\mathrm{ln}\left(e\right)=1$ 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 ${a}^{x}$.

## Practice set 1: exponent is $x$‍

Problem 1.1
$f\left(x\right)=-4{e}^{x}$
${f}^{\prime }\left(x\right)=?$

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

## Practice set 2: exponent is a polynomial

Problem 2.1
$y={e}^{\left(3{x}^{2}-4\right)}$
$\frac{dy}{dx}=?$

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

## Want to join the conversation?

• 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.
• 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).
• How can we differentiate e^x^x ? Or similar questions with double powers of exponential functions?
• 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.
(1 vote)
• Can someone help me with this question?

If f(x)=e^(2/x), then f'(x)=
(1 vote)
• is there an easy way to remember which kind of question goes to which type of answer?
• differentiate the following 7^3x+2
• 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)
(1 vote)
• Can someone help me to differentiate this : f (x)= xe^(-x^2/2)