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### Course: Calculus, all content (2017 edition) > Unit 2

Lesson 27: Exponential functions differentiation- Derivatives of sin(x), cos(x), tan(x), eˣ & ln(x)
- Derivative of aˣ (for any positive base a)
- Derivatives of aˣ and logₐx
- Worked example: Derivative of 7^(x²-x) using the chain rule
- Differentiate exponential functions
- Derivative of 2ˣ (old)
- Differentiating exponential functions review

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# 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:

Notice that ${e}^{x}$ is a specific case of the general form ${a}^{x}$ where $a=e$ . Since $\mathrm{ln}(e)=1$ 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 2: exponent is a polynomial

*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.(2 votes)
- 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).(3 votes)

- How can we differentiate e^x^x ? Or similar questions with double powers of exponential functions?(2 votes)
- 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)=(0 votes) - is there an easy way to remember which kind of question goes to which type of answer?(0 votes)
- Can someone help me to differentiate this : f (x)= xe^(-x^2/2)(0 votes)
- 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)(0 votes)

- differentiate the following 7^3x+2(0 votes)
- 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)(0 votes)