Derivative Problems with Detailed Solutions
This page presents carefully selected problems on
derivatives of functions, along with detailed solutions.
The goal is to strengthen both conceptual understanding and computational fluency in calculus.
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.
Solution
-
Since \( H(x) = f(x)g(x) \), we apply the product rule 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
\[
f(0) = 3,\quad f'(1) = 1,\quad f''(2) = 4.
\]
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 \( g'(2) \).
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 \( h'(10) \).
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