AP ECET MATHEMATICS Multiple Choice Questions (MCQs)
AP ECET Mathematics - Differentiation and its Applications
AP ECET Mathematics - Differentiation and its Applications
76. What is the first principle definition of the derivative of a function \( f(x) \)?
A) \( f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \)
B) \( f'(x) = \lim_{h \to \infty} \frac{f(x+h) - f(x)}{h} \)
C) \( f'(x) = \frac{f(x+h) + f(x)}{h} \)
D) \( f'(x) = \lim_{h \to 0} \frac{f(x) - f(x-h)}{h} \)
Answer: A) \( f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \)
77. The derivative of the sum of two functions \( f(x) \) and \( g(x) \) is given by:
A) \( f'(x) + g'(x) \)
B) \( f'(x) \cdot g'(x) \)
C) \( f'(x) - g'(x) \)
D) \( f'(x) \cdot g(x) \)
Answer: A) \( f'(x) + g'(x) \)
78. If \( f(x) = \sin x \), what is \( \frac{d}{dx} (\sin x) \)?
A) \( \cos x \)
B) \( -\cos x \)
C) \( \sin x \)
D) \( -\sin x \)
Answer: A) \( \cos x \)
79. The derivative of \( \tan x \) is:
A) \( \sec^2 x \)
B) \( \cosec^2 x \)
C) \( \cot^2 x \)
D) \( \sec x \)
Answer: A) \( \sec^2 x \)
80. What is the derivative of the function \( y = e^x \)?
A) \( e^x \)
B) \( x e^x \)
C) \( e^{x-1} \)
D) \( \ln x \)
Answer: A) \( e^x \)
81. What is the derivative of \( f(x) = \ln x \)?
A) \( \frac{1}{x} \)
B) \( \frac{1}{x^2} \)
C) \( \ln x \)
D) \( e^x \)
Answer: A) \( \frac{1}{x} \)
82. The second derivative of \( f(x) = x^3 \) is:
A) \( 6x \)
B) \( 3x^2 \)
C) \( 3x \)
D) \( 9x^2 \)
Answer: A) \( 6x \)
83. If \( y = x^2 \ln x \), what is \( \frac{dy}{dx} \)?
A) \( 2x \ln x + x \)
B) \( 2x \ln x + 2x \)
C) \( x \ln x + x^2 \)
D) \( 2x \ln x + x^2 \)
Answer: D) \( 2x \ln x + x^2 \)
84. What is the derivative of \( f(x) = \cos^{-1} x \)?
A) \( \frac{1}{\sqrt{1 - x^2}} \)
B) \( \frac{-1}{\sqrt{1 - x^2}} \)
C) \( \frac{1}{\sqrt{1 + x^2}} \)
D) \( \frac{-1}{\sqrt{1 + x^2}} \)
Answer: B) \( \frac{-1}{\sqrt{1 - x^2}} \)
85. Which of the following represents the derivative of \( y = \tan^{-1} x \)?
A) \( \frac{1}{1 + x^2} \)
B) \( \frac{1}{1 - x^2} \)
C) \( \frac{1}{1 + \sqrt{x^2}} \)
D) \( \frac{x}{1 + x^2} \)
Answer: A) \( \frac{1}{1 + x^2} \)
86. The derivative of \( f(x) = \ln(x^2 + 1) \) is:
A) \( \frac{2x}{x^2 + 1} \)
B) \( \frac{1}{x^2 + 1} \)
C) \( \frac{2}{x^2 + 1} \)
D) \( \frac{1}{x} \)
Answer: A) \( \frac{2x}{x^2 + 1} \)
87. If \( y = f(x) \cdot g(x) \), the product rule gives:
A) \( \frac{d}{dx}(f(x) \cdot g(x)) = f'(x) + g'(x) \)
B) \( \frac{d}{dx}(f(x) \cdot g(x)) = f'(x) \cdot g(x) + f(x) \cdot g'(x) \)
C) \( \frac{d}{dx}(f(x) \cdot g(x)) = f'(x) \cdot g'(x) \)
D) \( \frac{d}{dx}(f(x) \cdot g(x)) = f(x) + g(x) \)
Answer: B) \( \frac{d}{dx}(f(x) \cdot g(x)) = f'(x) \cdot g(x) + f(x) \cdot g'(x) \)
88. What is the derivative of \( f(x) = \sin(x^2) \)?
A) \( 2x \cos(x^2) \)
B) \( \cos(x^2) \)
C) \( 2x \sin(x^2) \)
D) \( 2x \)
Answer: A) \( 2x \cos(x^2) \)
89. The derivative of \( y = \frac{1}{x} \) is:
A) \( -\frac{1}{x^2} \)
B) \( \frac{1}{x^2} \)
C) \( -x \)
D) \( x^2 \)
Answer: A) \( -\frac{1}{x^2} \)
90. In which of the following cases is the derivative of \( y \) undefined?
A) \( y = |x| \)
B) \( y = x^3 \)
C) \( y = x^2 \)
D) \( y = \sin x \)
Answer: A) \( y = |x| \)
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