Differentiation Rules MCQs in Calculus

By: Prof. Dr. Fazal Rehman Shamil | Last updated: February 12, 2025

\[
\textbf{Question 1:}
\]
\[
\text{Which rule is used to differentiate the function } f(x) = g(x)h(x)?
\]
\[
\text{(a) } \text{Product Rule}, \quad \text{(b) } \text{Quotient Rule}, \quad \text{(c) } \text{Chain Rule}, \quad \text{(d) } \text{Power Rule}
\]
\[
\text{Answer: A}
\]

Step by Step Solution

\[
\textbf{Solution:}
\]
\[
\text{The Product Rule states that if } f(x) = g(x)h(x),
\]
\[
\text{then } f'(x) = g'(x)h(x) + g(x)h'(x).
\]
\[
\text{Thus, the correct answer is } \boxed{\text{Product Rule}}.
\]

\[
\textbf{Question 2:}
\]
\[
\text{Find the derivative of } f(x) = x^n \text{ where } n \text{ is a constant.}
\]
\[
\text{(a) } nx^{n-1}, \quad \text{(b) } n^x, \quad \text{(c) } x^n, \quad \text{(d) } n
\]
\[
\text{Answer: A}
\]

Step by Step Solution

\[
\textbf{Solution:}
\]
\[
\text{Using the Power Rule:} \quad \frac{d}{dx} x^n = nx^{n-1}.
\]
\[
\text{Thus, the correct answer is } \boxed{nx^{n-1}}.
\]

\[
\textbf{Question 3:}
\]
\[
\text{Which of the following is the derivative of } f(x) = \sin(x)?
\]

\[
\text{(a) } -\cos(x), \quad \text{(b) } \sin(x), \quad \text{(c) } \cos(x), \quad \text{(d) } -\sin(x)
\]

\[
\text{Answer: c}
\]

Step by Step Solution

\[
\textbf{Solution:}
\]
\[
\text{The derivative of } \sin(x) \text{ is } \cos(x).
\]
\[
\frac{d}{dx} \sin(x) = \cos(x).
\]
\[
\text{Thus, the correct answer is } \boxed{\cos(x)}.
\]

\[
\textbf{Question 4:}
\]
\[
\text{The derivative of } f(x) = \ln(x) \text{ is:}
\]
\[
\text{(a) } \frac{1}{x}, \quad \text{(b) } \ln(x), \quad \text{(c) } x, \quad \text{(d) } x^2
\]
\[
\text{Answer: A}
\]

Step by Step Solution

\[
\textbf{Solution:}
\]
\[
\text{The derivative of } \ln(x) \text{ is } \frac{1}{x}.
\]
\[
\frac{d}{dx} \ln(x) = \frac{1}{x}.
\]
\[
\text{Thus, the correct answer is } \boxed{\frac{1}{x}}.
\]

\[
\textbf{Question 5:}
\]
\[
\text{Find the derivative of } f(x) = e^x.
\]

\[
\text{(a) } 1, \quad \text{(b) } x, \quad \text{(c) } e^x, \quad \text{(d) } x e^x
\]

\[
\text{Answer: c}
\]

Step by Step Solution

\[
\textbf{Solution:}
\]
\[
\text{The derivative of } e^x \text{ is } e^x.
\]
\[
\frac{d}{dx} e^x = e^x.
\]
\[
\text{Thus, the correct answer is } \boxed{e^x}.
\]

\[
\textbf{Question 6:}
\]
\[
\text{What is the derivative of } f(x) = \tan(x)?
\]
\[
\text{(a) } \sec^2(x), \quad \text{(b) } \sin(x), \quad \text{(c) } \cos(x), \quad \text{(d) } \sec(x)
\]
\[
\text{Answer: A}
\]

Step by Step Solution

\[
\textbf{Solution:}
\]
\[
\text{The derivative of } \tan(x) \text{ is } \sec^2(x).
\]
\[
\frac{d}{dx} \tan(x) = \sec^2(x).
\]
\[
\text{Thus, the correct answer is } \boxed{\sec^2(x)}.
\]

\[
\textbf{Question 7:}
\]
\[
\text{Find the derivative of } f(x) = \frac{1}{x}.
\]

\[
\text{(a) } -x, \quad \text{(b) } x, \quad \text{(c) } -\frac{1}{x^2}, \quad \text{(d) } \frac{1}{x^2}
\]

\[
\text{Answer: c}
\]

Step by Step Solution

\[
\textbf{Solution:}
\]
\[
\text{Rewrite the function as } f(x) = x^{-1}.
\]
\[
\text{Differentiate using the Power Rule:} \quad \frac{d}{dx} x^n = n x^{n-1}.
\]
\[
\frac{d}{dx} x^{-1} = -1 \cdot x^{-2} = -\frac{1}{x^2}.
\]
\[
\text{Thus, the correct answer is } \boxed{-\frac{1}{x^2}}.
\]

\[
\textbf{Question 8:}
\]
\[
\text{Which rule is used to differentiate the composite function } f(x) = g(h(x))?
\]
\[
\text{(a) } \text{Product Rule}, \quad \text{(b) } \text{Chain Rule}, \quad \text{(c) } \text{Quotient Rule}, \quad \text{(d) } \text{Power Rule}
\]
\[
\text{Answer: B}
\]

Step by Step Solution

\[
\textbf{Solution:}
\]
\[
\text{The Chain Rule states that if } f(x) = g(h(x)),
\]
\[
\text{then } f'(x) = g'(h(x)) \cdot h'(x).
\]
\[
\text{Thus, the correct answer is } \boxed{\text{Chain Rule}}.
\]

\[
\textbf{Question 9:}
\]
\[
\text{The derivative of } f(x) = \cos(x) \text{ is:}
\]
\[
\text{(a) } -\sin(x), \quad \text{(b) } \sin(x), \quad \text{(c) } -\cos(x), \quad \text{(d) } \cos(x)
\]
\[
\text{Answer: A}
\]

Step by Step Solution

\[
\textbf{Solution:}
\]
\[
\text{The derivative of } \cos(x) \text{ is } -\sin(x).
\]
\[
\frac{d}{dx} \cos(x) = -\sin(x).
\]
\[
\text{Thus, the correct answer is } \boxed{-\sin(x)}.
\]

\[
\textbf{Question 10:}
\]
\[
\text{If } f(x) = x^2 + 3x + 5, \text{ the derivative is:}
\]
\[
\text{(a) } 2x + 3, \quad \text{(b) } 2x + 5, \quad \text{(c) } 2x, \quad \text{(d) } 3x + 5
\]
\[
\text{Answer: A}
\]

Step by Step Solution

\[
\textbf{Solution:}
\]
\[
\frac{d}{dx} (x^2 + 3x + 5) = 2x + 3.
\]
\[
\text{Thus, the correct answer is } \boxed{2x + 3}.
\]

\[
\textbf{Question 11:}
\]
\[
\text{Which of the following is the derivative of } f(x) = \sec(x)?
\]
\[
\text{(a) } \sec(x)\tan(x), \quad \text{(b) } \sec(x), \quad \text{(c) } \tan(x), \quad \text{(d) } -\sec(x)
\]
\[
\text{Answer: A}
\]

Step by Step Solution

\[
\textbf{Solution:}
\]
\[
\frac{d}{dx} \sec(x) = \sec(x) \tan(x).
\]
\[
\text{Thus, the correct answer is } \boxed{\sec(x)\tan(x)}.
\]

\[
\textbf{Question 12:}
\]
\[
\text{The derivative of } f(x) = \arcsin(x) \text{ is:}
\]
\[
\text{(a) } \frac{1}{\sqrt{1 – x^2}}, \quad \text{(b) } \frac{1}{x^2}, \quad \text{(c) } \frac{1}{x\sqrt{1 – x^2}}, \quad \text{(d) } \sqrt{1 – x^2}
\]
\[
\text{Answer: A}
\]

Step by Step Solution

\[
\textbf{Solution:}
\]
\[
\frac{d}{dx} \arcsin(x) = \frac{1}{\sqrt{1 – x^2}}.
\]
\[
\text{Thus, the correct answer is } \boxed{\frac{1}{\sqrt{1 – x^2}}}.
\]

\[
\textbf{Question 13:}
\]
\[
\text{Find the derivative of } f(x) = \log_a(x).
\]

\[
\text{(a) } x \ln(a), \quad \text{(b) } \frac{1}{x}, \quad \text{(c) } \ln(a), \quad \text{(d) } \frac{1}{x \ln(a)}
\]

\[
\text{Answer: d}
\]

Step by Step Solution

\[
\textbf{Solution:}
\]
\[
\frac{d}{dx} \log_a(x) = \frac{1}{x \ln(a)}.
\]
\[
\text{Thus, the correct answer is } \boxed{\frac{1}{x \ln(a)}}.
\]

\[
\textbf{Question 14:}
\]
\[
\text{What is the derivative of } f(x) = \sqrt{x}?
\]
\[
\text{(a) } \frac{1}{2\sqrt{x}}, \quad \text{(b) } \frac{1}{\sqrt{x}}, \quad \text{(c) } \frac{1}{2x}, \quad \text{(d) } x^{\frac{1}{2}}
\]
\[
\text{Answer: A}
\]

Step by Step Solution

\[
\textbf{Solution:}
\]
\[
\text{Rewrite as } f(x) = x^{\frac{1}{2}}.
\]
\[
\frac{d}{dx} x^{\frac{1}{2}} = \frac{1}{2} x^{-\frac{1}{2}} = \frac{1}{2\sqrt{x}}.
\]
\[
\text{Thus, the correct answer is } \boxed{\frac{1}{2\sqrt{x}}}.
\]

\[
\textbf{Question 15:}
\]
\[
\text{If } f(x) = \frac{2x^2 + 3x}{x^2 + 1}, \text{ what is its derivative?}
\]
\[
\text{(a) } \frac{(4x + 3)(x^2 + 1) – (2x^2 + 3x)(2x)}{(x^2 + 1)^2}, \quad \text{(b) } \frac{4x + 3}{x^2 + 1}, \quad \text{(c) } \frac{2x^2 + 3x}{x}, \quad \text{(d) } \frac{4x + 3}{x}
\]
\[
\text{Answer: A}
\]

Step by Step Solution

\[
\textbf{Solution:}
\]
\[
\text{Using the Quotient Rule: } \left( \frac{g(x)}{h(x)} \right)’ = \frac{g'(x)h(x) – g(x)h'(x)}{h(x)^2}.
\]
\[
\text{Thus, the correct answer is } \boxed{\frac{(4x + 3)(x^2 + 1) – (2x^2 + 3x)(2x)}{(x^2 + 1)^2}}.
\]

More MCQs on Calculus

  1. Functions and Models MCQs in Calculus
  2. Limits and Derivatives MCQs in Calculus
  3. Differentiation Rules MCQs in Calculus
  4. Applications of Differentiation MCQs in Calculus
  5. Integrals MCQs in Calculus
  6. Applications of Integration MCQs in Calculus
  7. Techniques of Integration MCQs in Calculus
  8. Differential Equations MCQs in Calculus
  9. Parametric Equations and Polar Coordinates MCQs in Calculus
  10. Infinite Sequences and Series MCQs in Calculus
  11. Vectors and the Geometry of Space MCQs in Calculus
  12. Vector Functions MCQs in Calculus
  13. Partial Derivatives MCQs in Calculus
  14. Multiple Integrals MCQs in Calculus
  15. Vector Calculus MCQs in Calculus
  16. Second-Order Differential Equations MCQs in Calculus

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