2.8.3: Additional Exercises

Reading Questions

1. What is the derivative of \(f(x) = \ln x\)?
2. How is implicit differentiation used to find the derivative of \(y = \ln x\)?
3. What is the derivative of \(f(x) = \log_b x\)? How does it relate to the derivative of \(\ln x\)?
4. Why is it often beneficial to use properties of logarithms before differentiating a function involving logarithms?
5. What is the primary purpose of logarithmic differentiation? For what types of functions is it particularly useful?
6. What is the first step in performing logarithmic differentiation for a function \(y=f(x)\)?
7. Which properties of logarithms are commonly used to simplify the expression after taking the natural logarithm of both sides? Give an example for \(\ln(A \cdot B)\), \(\ln(A/B)\), and \(\ln(A^n)\).
8. After simplifying \(\ln y\), what differentiation technique is applied to both sides of the equation with respect to \(x\)?
9. When differentiating \(\ln y\) with respect to \(x\), what is the result?
10. After differentiating both sides and obtaining an expression for \(\frac{1}{y}\frac{dy}{dx}\), what is the final step to find \(\frac{dy}{dx}\)?
11. For the function \(y = (2x^4+1)^{\tan x}\), what does \(\ln y\) become after applying logarithm properties?
12. Can logarithmic differentiation be used for functions that could also be differentiated by the Product or Quotient Rule? What might be an advantage of using it in such cases?
13. When would you choose to use logarithmic differentiation over other known rules?

Homework

In exercises 1 - 12, find the derivatives for the functions.

1) \(f(x) = \sinh(\ln(x))\)

2) \( \cos{(\ln x)} \)

3) \(f(x)=\ln(4x^3+x)\)

4) \(f(x)=\log(\sec x)\)

Answer

\(f^{\prime}(x) = \dfrac{\tan x}{\ln 10}\)

5) \(\ln(\tanh^{−1}(x))\)

Answer

\(−\dfrac{1}{(x^2−1)\tanh^{−1}(x)}\)

6) \(f(x) = \ln(\text{sech}(x)+\tanh(x))\)

7) \(f(x)=e^{x^3\ln x}\)

Answer

\(f^{\prime}(x) = e^{x^3\ln x}\left(3x^2\ln x+x^2\right)\)

8) \(f(x)=x^2\ln 9x\)

9) \(f(x)=2^x \cdot \log_37^{x^2−4}\)

Answer

\(f^{\prime}(x) = 2^x \cdot \ln 2 \cdot \log_3 7^{x^2−4}+2^x \cdot \dfrac{2x\ln 7}{\ln 3}\)

10) \( \ln{\left( \cos^2 x\right)} \)

11) \(f(x)=\ln\sqrt{5x−7}\)

12) \(f(x)=\log_7(6x^4+3)^5\)

Answer

\(f^{\prime}(x) = \dfrac{5}{2(5x−7)}\)

13) [Technology Required] Find the equation of the tangent line to the graph of \(x^3−x\ln y+y^3=2x+5\) at the point where \(x=2\). (Hint: Use implicit differentiation to find \(\dfrac{dy}{dx}\).) Graph both the curve and the tangent line.

For exercises 14 - 21, use logarithmic differentiation to find \(\dfrac{dy}{dx}\).

14) \(y=x^{\sqrt{x}}\)

15) \(y=(\sin 2x)^{4x}\)

Answer

\(\dfrac{dy}{dx} = (\sin 2x)^{4x}\big[4 \cdot \ln(\sin 2x)+8x \cdot \cot 2x\big]\)

16) \(y=(\ln x)^{\ln x}\)

17) \(y=x^{\log_2x}\)

Answer

\(\dfrac{dy}{dx} = x^{\log_2x} \cdot \dfrac{2\ln x}{x\ln 2}\)

18) \(y=(x^2−1)^{\ln x}\)

19) \(y=x^{\cot x}\)

Answer

\(\dfrac{dy}{dx} = x^{\cot x} \cdot \left[−\csc^2x \cdot \ln x+\dfrac{\cot x}{x}\right]\)

20) \(y=\dfrac{x+11}{\sqrt[3]{x^2−4}}\)

21) \(y=x^{−1/2}(x^2+3)^{2/3}(3x−4)^4\)

Answer

\(\dfrac{dy}{dx} = x^{−1/2}(x^2+3)^{2/3}(3x−4)^4 \cdot \left[\dfrac{−1}{2x}+\dfrac{4x}{3(x^2+3)}+\dfrac{12}{3x−4}\right]\)

22) Consider the function \(y=x^{1/x}\) for \(x>0\).

a. Determine the points on the graph where the tangent line is horizontal.

b. Determine the points on the graph where \(y^{\prime}>0\) and those where \(y^{\prime}<0\).

Answer

a. \(x=e \approx 2.718\)
b. \(y^{\prime}>0 \text{ for } (0,e)\) and \(y^{\prime}<0 \text{ for } (e, \infty )\).