Sec 3.2: Product and Quotient Rules

Product Rule (formula derivation)
  1. Find the derivatives of the following functions. You don't have to simplify your answer. solution
    (a) \( f(x) = (x^2 + 2x)(3x - 5) \)   (b) \( f(x) = (5x - 3x^5 - x^7)(3x^2 - 5x) \)
  2. Find the derivatives of the following functions. You don't have to simplify your answer. solution
    (a) \( f(t) = e^t(t^3 + 2t) \)   (b) \( g(t) = \dfrac{e^t - 1}{5t^3 + 7t} \)
  3. Find the derivatives of the following functions. You don't have to simplify your answer. solution
    (a) \( f(x) = \dfrac{x^2 + 2x}{5x - 1} \)   (b) \( f(x) = \dfrac{7 + 3x^5 - x^7}{3x^2 - 5x} \)
  4. Differentiate \( \displaystyle f(x) = \dfrac{2 - x e^{x}}{x + e^{x}} \). solution
  5. If \( \displaystyle f(x) = \dfrac{x^2 + 3x + 1}{2x - 5} \), find the value of \( f'(0) \). solution
  6. If \( F(x) = (x^2 + 1)e^x \), find \( F'(0) \), and if \( \displaystyle G(x) = \dfrac{x}{x - 3} \), find \( G'(-1) \). solution
  7. If \( g(x) = \sqrt{\dfrac{1}{x}} \), find the value of \( g'(4) \). solution
  8. Find the first and second derivative of \( f(x) = e^x \sqrt{x} \). solution
  9. Find an equation of the tangent line to the curve \( \displaystyle y = \dfrac{e^x}{1 + x^2} \) at the point \( \left(1, \dfrac{1}{2}e \right) \). solution
  10. Refer the table for the questions (10)–(12) given below. solution
    \( x \) \( f(x) \) \( f'(x) \) \( g(x) \) \( g'(x) \)
    0 2 -2 4 1
    1 2 3 1 2
    1. If \( H(x) = 2f(x) - g(x) \), find \( H'(0) \).
    2. If \( J(x) = f(x)g(x) \), find \( J'(1) \).
    3. If \( K(x) = e^x f(x) \), find \( K'(0) \).
  11. Suppose that \( f(5) = 3 \), \( f'(5) = 2 \), \( g(5) = -6 \), and \( g'(5) = 7 \). Find the following values: (a) \( (fg)'(5) \), (b) \( (f/g)'(5) \), (c) \( (g/f)'(5) \). solution
  12. If \( F(x) = x e^x \), find (i) \( F'(\ln 2) \) and (ii) \( F''(0) \). solution