Derivative Problems with Detailed Solutions

This page presents carefully selected problems on derivatives of functions[cite: 1]. Each solution is hidden inside a collapsible dropdown so you can practice independently before reviewing the steps.

Questions and Solutions

Question 1

Let \( f \), \( g \), and \( H \) be functions defined such that \[ H(x) = (f g)(x) \] and \[ f(1) = 36,\quad f'(-2) = 3,\quad f'(1) = 4,\quad g(1) = 9,\quad g'(1) = -1. \] Determine whether the slope of the tangent line to the graph of \( H \) at \( x = 1 \) is positive, negative, or zero.

View Solution
  • Since \( H(x) = f(x)g(x) \), we apply the product rule for its derivative: \[ H'(x) = f'(x)g(x) + f(x)g'(x) \]
  • Evaluate at \( x = 1 \): \[ H'(1) = f'(1)g(1) + f(1)g'(1) \]
  • Substitute the given values: \[ H'(1) = (4)(9) + (36)(-1) = 36 - 36 = 0 \]
  • Since the derivative at \( x = 1 \) equals zero, the slope of the tangent line is zero, and the tangent line is parallel to the \( x \)-axis.

Question 2

Let \[ f(x) = ax^2 + bx + c. \] Find the values of \( a \), \( b \), and \( c \) such that \[ f(0) = 3,\quad f'(1) = 1,\quad f''(2) = 4. \]

View Solution
  • Using \( f(0) = 3 \): \[ a(0)^2 + b(0) + c = 3 \Rightarrow c = 3 \]
  • Compute the derivatives: \[ f'(x) = 2ax + b,\qquad f''(x) = 2a \]
  • Use \( f''(2) = 4 \): \[ 2a = 4 \Rightarrow a = 2 \]
  • Use \( f'(1) = 1 \): \[ 2(2)(1) + b = 1 \Rightarrow b = -3 \]
  • The solution is: \[ a = 2,\quad b = -3,\quad c = 3 \]

Question 3

Let \( f(x) = x^3 + x \) and let \( g(x) = f^{-1}(x) \). Find the value of \( g'(2) \).

View Solution
  • Since \( g(x) = f^{-1}(x) \), we have: \[ f(g(x)) = x \]
  • Differentiate both sides using the chain rule: \[ f'(g(x))\,g'(x) = 1 \]
  • Evaluate at \( x = 2 \): \[ f'(g(2))\,g'(2) = 1 \]
  • Since \( f(1) = 2 \), it follows that \( g(2) = 1 \)
  • Compute \( f'(x) = 3x^2 + 1 \), hence: \[ f'(1) = 4 \]
  • Therefore: \[ g'(2) = \frac{1}{4} \]

Question 4

Let \( g(x) = f^{-1}(x) \) and \( h(x) = (g(x))^5 \). Given that \[ f(6) = 10,\quad f'(6) = 12, \] find \( h'(10) \).

View Solution
  • Differentiate \( h(x) \): \[ h'(x) = 5g'(x)g(x)^4 \]
  • Evaluate at \( x = 10 \): \[ h'(10) = 5g'(10)g(10)^4 \]
  • Since \( g(10) = f^{-1}(10) = 6 \)
  • Differentiate \( f(g(x)) = x \): \[ f'(g(x))g'(x) = 1 \]
  • Evaluate at \( x = 10 \): \[ f'(6)g'(10) = 1 \Rightarrow g'(10) = \frac{1}{12} \]
  • Compute: \[ h'(10) = 5\left(\frac{1}{12}\right)6^4 = 540 \]

More References on Calculus