Section 3.6: Derivative of Logarithmic functions

Differentiate the following functions: 1--8.

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


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


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


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


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


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


8. (a) \(\displaystyle{f(x) = \log_3 (x^2+1)}\)    (b) \(\displaystyle{f(x) = \log_5(x \tan x)}\)   solution


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


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


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

 

12. Find an equation of the tangent line to the curve   \( y=\ln(x^3-7) \)   at the point \( (2, 0) \).   solution 


13. Find an equation of the tangent line to the curve  \( \displaystyle y=\frac{5\ln(x)}{x} \)   at the point \( (e, 5/e) \).     solution


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

 

Logarithmic Differentiation:

Use logarithmic differentiation to find the derivative of the function.

(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