Exponential and Log Derivatives Practice Problems with Answers

These derivatives of exponential and logarithmic functions practice problems start with the basic rules, then use the chain rule, log properties and logarithmic differentiation. Try each one first. The answers come right after the list, then the full solutions.

The rules you need:

With an inside function u: (e^u)' = e^u u' and (\ln u)' = \frac{u'}{u}. See the derivative of e^x, the derivative of ln x and the derivative of 2^x for the proofs.

Problems

Basic rules

1. y = 5e^x - \ln x
2. y = 2^x
3. y = \log_{10} x

Chain rule

4. y = e^{3x}
5. y = \ln(x^2 + 1)
6. y = 5^{x^2}

Use log rules first

7. y = \ln(5x)
8. y = \ln\sqrt{x^2 + 4}
9. y = \ln\!\left(\dfrac{x + 1}{x - 1}\right)

With other rules

10. y = x^2\ln x
11. y = \dfrac{e^x}{1 + e^x}
12. y = x^x for x > 0

Answers

Step-by-step solutions

1. Differentiate each term:

2. Use (a^x)' = a^x\ln a with a = 2:

3. Use (\log_a x)' = \frac{1}{x\ln a} with a = 10, or write \log_{10} x = \frac{\ln x}{\ln 10} first:

4. The inside is 3x, with derivative 3:

5. Use (\ln u)' = \frac{u'}{u} with u = x^2 + 1:

6. Use (a^u)' = a^u\ln a\cdot u' with a = 5 and u = x^2:

7. Split the log: \ln(5x) = \ln 5 + \ln x. The first term is a constant:

The chain rule gives the same thing: \frac{5}{5x} = \frac{1}{x}.

8. A square root is a power of \frac{1}{2}, and logs bring powers to the front:

9. Split the quotient: y = \ln(x + 1) - \ln(x - 1). Then

This is much shorter than the quotient rule inside the chain rule.

10. Use the product rule:

11. Use the quotient rule. Both the top and the bottom have derivative e^x:

12. Neither the power rule nor the exponential rule fits, because x is in both places. Use logarithmic differentiation: take \ln of both sides.

Differentiate both sides, using the product rule on the right:

Common mistakes on these problems

• Using the power rule on e^x or 2^x. (2^x)' is not x\cdot 2^{x-1}. The variable is in the exponent, so you need 2^x\ln 2.
• Writing \ln(x^2 + 1)' = \frac{1}{x^2 + 1}. That misses the inside derivative 2x on top.
• Splitting a log the wrong way. \ln(x + 1) \ne \ln x + \ln 1. Log rules split products and quotients, not sums.
• Treating x^x as x\cdot x^{x-1}. Problem 12 needs logarithmic differentiation. See the derivative of x^x for more.

Keep practicing

These 12 problems use fixed numbers. For unlimited practice with new functions and instant feedback, try the exponential and log derivatives lesson, then the chain rule with exponentials lesson. For more worked examples, read the exponential and log derivatives guide.