Section 3.6: Derivative of Logarithmic Functions

1. (a) \( f(x) = e^{-5x} \)   (b) \( f(x) = (2 + \ln x)^5 \)   (c) \( f(x) = \ln(5x^2 - x + 3) \)  solution

2. (a) \( f(x) = \sqrt[4]{\ln x} \)   (b) \( f(x) = \ln(x^3 + 7) \)   (c) \( g(x) = 2x\ln(3x) - 5x \)  solution

3. (a) \( f(x) = \ln(\ln x) \)   (b) \( f(t) = \ln(\ln(5t)) \)   (c) \( g(t) = \ln(te^{-2t}) \)  solution

4. (a) \( f(x) = \ln(\sin^2 x) \)   (b) \( f(t) = \ln(\sqrt{\ln(2t + 3)}) \)  solution

5. (a) \( f(x) = \frac{\ln x}{e^x - x} \)   solution   (b) \( f(x) = \ln(x + \sqrt{x^2 - 2}) \)  solution

6. (a) \( f(x) = \tan(\ln(2x)) \)   (b) \( f(x) = \cos\left(\frac{x}{\ln x}\right) \)  solution

7. (a) \( f(x) = \ln\left(\frac{2x}{x - 1}\right) \)   (b) \( f(x) = \ln(x^2 e^x) \)  solution

8. (a) \( f(x) = \log_{10}(\sqrt{x}) \)   (b) \( f(x) = \log_5(x^3 - 1) \)  solution

9. (a) \( f(x) = \log_3(x^2 + 1) \)   (b) \( f(x) = \log_5(x\tan x) \)  solution

10. Differentiate two different ways: \( f(x) = \ln(xe^{-9x}) \)  solution

11. If \( f(x) = \frac{\ln x}{x^2} \), find \( f'(1) \).  solution

12. If \( f(x) = \ln(x^2 - 3x + 1) \), find \( f'(0) \).  solution

13. Find the tangent line to \( y = \ln(x^3 - 7) \) at \( (2,0) \).  solution

14. Find the tangent line to \( y = \frac{5\ln x}{x} \) at \( (e, 5/e) \).  solution

15. Use logarithmic differentiation to find the derivative of \( y = (x^3 + 2)^2 (x^5 + 3)^4 \).  solution

Logarithmic Differentiation

Use logarithmic differentiation to find the derivative.

(a) \( y = x^x \)  solution   (b) \( y = (x^2 + 1)^x \)  solution   (c) \( y = (5 + 2x)^x \)  solution

(d) \( y = x^{\sin x} \)  solution   (e) \( y = (\tan x)^x \)  solution