BC Calculus: Assignment 48: Review for Chapter 6 Test Worksheet

In the following problems, find \(f'(x)\).
  1. \(f(x) = \ln{(x^4 + 8)} \)
    2. Solution \( f'(x) = \frac{4x^3}{x^4 + 8} \)
  2. \(f(x) = \ln{(3x\sqrt{3 - x})} \)
    3. Solution \( f(x) = \ln{3} + \ln{x} + \frac{1}{2} \ln{(3 - x)} \implies f'(x) = \frac{1}{x} - \frac{1}{2(3 - x)} \)
  3. \( f(x) = \ln{ \left( \frac{5x^2}{\sqrt{5 + x^2}} \right) } \)
    4. Solution \( f(x) = \ln{5} + 2\ln{x} - \frac{1}{2} \ln{(5 + x^2)} \implies f'(x) = \frac{2}{x} - \frac{2x}{2(5 + x^2)} \)
  4. \( f(x) = e^{x \cos{x}} \)
    5. Solution \( f'(x) = e^{x \cos{x}} ( \cos{x} - x \sin{x}) \)
  5. \( f(x) = e^{-3x}\sin{(5x)}\)
    6. Solution \( f'(x) = e^{-3x}\cos{(5x)}5 + \sin{(5x)} e^{-3x}(-3) \)
  6. \( f(x) = e^{\ln{\frac{1}{x}}}\)
    7. Solution \( f(x) = \frac{1}{x} = x^{-1} \implies f'(x) = -x^{-2} \)
  7. \( f(x) = log_{12} { \left( x^3 \right) }\)
    8. Solution \( f'(x) = \frac{3}{x \ln{12}} \)
  8. \( f(x) = x^5 5^x\)
    9. Solution \( f'(x) = x^55^x\ln{5} + 5^x5x^4 \)
Evaluate the following integrals:
  1. \(\displaystyle \int\frac{\sec^2{x}}{\tan{x}} \, dx \)
    2. Solution \( \ln{|\tan{x}|} + C \)
  2. \(\displaystyle \int\frac{\cos{x}}{1 - \sin{x}} \, dx \)
    3. Solution \( -\ln{|1 - \sin{x}|} + C \)
  3. \(\displaystyle \int\frac{1}{x \ln{x}} \, dx \)
    4. Solution \( \ln {|\ln{x}|} + C \)
  4. \(\displaystyle \int\frac{1}{x} \cos{( \ln{x} )} \, dx \)
    5. Solution \( \sin{(\ln{x})} + C \)
  5. \(\displaystyle \int\frac{\sin{x} - \cos{x}}{\cos{x}} \, dx \)
    6. Solution \( \ln{|\sec{x}|} - x + C \)
  6. \(\displaystyle \int\frac{dx}{\sqrt{x} \left( 1 + 2\sqrt{x} \right)} \)
    7. Solution \( \ln{|1 + 2\sqrt{x}|} + C \)
  7. \(\displaystyle \int x \cdot 4^{-x^2} \, dx \)
    8. Solution \( -\frac{4^{-x^2}}{2\ln{4}} + C \)
  8. \(\displaystyle \int 7^{\sin{x}}\cos{x} \, dx \)
    9. Solution \( \frac{7^{\sin{x}}}{\ln{7}} + C \)