Differentiation MCQ

Ultimate Differentiation Challenge: Master the Derivatives!

Time Left: 45:00

Total Marks: 60, Obtained Marks: 0

1. What is the derivative of \( 3x^3 \)?

A. \( x^3 \)

B. \( 9x^2 \)

C. \( 3x^2 \)

D. \( 6x \)

The correct answer is \( 9x^2 \). The derivative of \( ax^n \) is \( a \cdot n x^{n-1} \), so for \( 3x^3 \), it is \( 9x^2 \).

2. What is the derivative of a constant \( c \)?

A. \( 0 \)

B. \( c \)

C. \( 1 \)

D. \( cx \)

The derivative of any constant is \( 0 \), as constants do not change.

3. Find the derivative of \( (5x^2 + 1)^3 \).

A. \( (5x^2 + 1)^2 \)

B. \( 30x(5x^2 + 1)^2 \)

C. \( 3(5x^2 + 1)^2 \)

D. \( 15x^2 \)

By the chain rule, the derivative of \( (5x^2 + 1)^3 \) is \( 3(5x^2 + 1)^2 \cdot (10x) \), which simplifies to \( 30x(5x^2 + 1)^2 \).

4. What is the derivative of \( x \cdot e^x \)?

A. \( e^x + xe^x \)

B. \( e^x \)

C. \( xe^x \)

D. \( x \)

Using the product rule, the derivative of \( x \cdot e^x \) is \( e^x + xe^x \).

5. What is the derivative of \( \sin(x) \)?

A. \( \sin(x) \)

B. \( \cos(x) \)

C. \( -\cos(x) \)

D. \( -\sin(x) \)

The derivative of \( \sin(x) \) is \( \cos(x) \).

6. What is the derivative of \( e^{2x} \)?

A. \( e^{2x} \)

B. \( 2e^x \)

C. \( 2e^{2x} \)

D. \( x e^{2x} \)

The derivative of \( e^{2x} \) is \( 2e^{2x} \), using the chain rule.

7. What is the derivative of \( \ln(x) \)?

A. \( \frac{1}{x} \)

B. \( x \ln(x) \)

C. \( \ln(x) \)

D. \( \frac{1}{x^2} \)

The derivative of \( \ln(x) \) is \( \frac{1}{x} \).

8. What is the derivative of \( \cos(x) \)?

A. \( \sin(x) \)

B. \( -\cos(x) \)

C. \( -\sin(x) \)

D. \( \cos(x) \)

The derivative of \( \cos(x) \) is \( -\sin(x) \).

9. What is the derivative of \( \tan(x) \)?

A. \( \sec^2(x) \)

B. \( \sec(x) \)

C. \( \csc(x) \)

D. \( \cot(x) \)

The derivative of \( \tan(x) \) is \( \sec^2(x) \).

10. What is the derivative of \( \sqrt{x} \)?

A. \( \frac{1}{2\sqrt{x}} \)

B. \( \frac{1}{2\sqrt{x}} \)

C. \( \frac{1}{x} \)

D. \( \sqrt{x} \)

The derivative of \( \sqrt{x} \) is \( \frac{1}{2\sqrt{x}} \), using the power rule.

11. Find the derivative of \( x^x \).

A. \( x^x \ln(x) \)

B. \( x^x (\ln(x) + 1) \)

C. \( x^x \cdot \ln(x) \)

D. \( x^x + \ln(x) \)

The derivative of \( x^x \) is \( x^x (\ln(x) + 1) \), using logarithmic differentiation.

12. What is the derivative of \( \frac{1}{x} \)?

A. \( -\frac{1}{x^2} \)

B. \( \frac{1}{x^2} \)

C. \( \frac{-1}{x} \)

D. \( \frac{1}{x} \)

The derivative of \( \frac{1}{x} \) is \( -\frac{1}{x^2} \), using the power rule.

13. What is the derivative of \( \sin^2(x) \)?

A. \( 2\sin(x)\cos(x) \)

B. \( 2\sin(x)\cos(x) \)

C. \( \cos^2(x) \)

D. \( \sin(x) \)

The derivative of \( \sin^2(x) \) is \( 2\sin(x)\cos(x) \), using the chain rule.

14. What is the derivative of \( \frac{e^x}{x} \)?

A. \( \frac{e^x (x - 1)}{x^2} \)

B. \( \frac{e^x}{x^2} \)

C. \( \frac{e^x}{x} \)

D. \( e^x \ln(x) \)

The derivative of \( \frac{e^x}{x} \) is \( \frac{e^x (x - 1)}{x^2} \), using the quotient rule.

15. What is the derivative of \( \sec(x) \)?

A. \( \sec(x) \tan(x) \)

B. \( \sec(x) \tan(x) \)

C. \( \sec^2(x) \)

D. \( \sec(x) \)

The derivative of \( \sec(x) \) is \( \sec(x) \tan(x) \).