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## Calculus, all content (2017 edition)

### 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, start superscript, x, end superscript is a specific case of the general form a, start superscript, x, end superscript where a, equals, e. Since natural log, left parenthesis, e, right parenthesis, equals, 1 we obtain the same result.

You can actually use the derivative of e, start superscript, x, end superscript (along with the chain rule) to obtain the general derivative of a, start superscript, x, end superscript.

*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)=(1 vote) - is there an easy way to remember which kind of question goes to which type of answer?(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)(1 vote)

- 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)(1 vote)