Q.1 If f(x) = x^3 + 2x^2 - x + 5, what is f'(x)?
3x^2 + 2x - 1
3x^2 + 4x - 1
x^2 + 4x - 1
3x^2 + 2x + 1
Explanation - Derivative of x^3 is 3x^2, derivative of 2x^2 is 4x, derivative of -x is -1, and derivative of 5 is 0. Adding them gives 3x^2 + 4x - 1.
Correct answer is: 3x^2 + 4x - 1
Q.2 Evaluate the derivative of f(x) = sin(x) + cos(x).
cos(x) - sin(x)
-cos(x) + sin(x)
cos(x) + sin(x)
-cos(x) - sin(x)
Explanation - Derivative of sin(x) is cos(x), derivative of cos(x) is -sin(x), so f'(x) = cos(x) - sin(x).
Correct answer is: cos(x) - sin(x)
Q.3 Find the integral ∫(2x + 3) dx.
x^2 + 3x + C
x^2 + 3 + C
2x^2 + 3x + C
x^2 + 3x
Explanation - Integral of 2x is x^2, integral of 3 is 3x, and add constant C: ∫(2x+3) dx = x^2 + 3x + C.
Correct answer is: x^2 + 3x + C
Q.4 If f(x) = e^x * x^2, find f'(x).
2x e^x
x^2 e^x + 2x e^x
x^2 e^x
2x e^(2x)
Explanation - Use the product rule: (u*v)' = u'v + uv'. Here, u = x^2, v = e^x. So derivative is 2x*e^x + x^2*e^x = x^2 e^x + 2x e^x.
Correct answer is: x^2 e^x + 2x e^x
Q.5 Compute the second derivative of f(x) = x^3.
6x
3x^2
6x^2
3x
Explanation - First derivative: f'(x) = 3x^2. Second derivative: f''(x) = 6x.
Correct answer is: 6x
Q.6 Evaluate ∫_0^1 x dx.
1/2
1
0
2
Explanation - Integral of x is x^2/2. Evaluate from 0 to 1: (1^2/2) - (0^2/2) = 1/2.
Correct answer is: 1/2
Q.7 Find the derivative of f(x) = ln(x^2 + 1).
2x / (x^2 + 1)
1 / (x^2 + 1)
ln(2x) / (x^2 + 1)
2 / (x^2 + 1)
Explanation - Derivative of ln(u) is u'/u. Here u = x^2 + 1, u' = 2x, so f'(x) = 2x / (x^2 + 1).
Correct answer is: 2x / (x^2 + 1)
Q.8 Find ∫ e^(3x) dx.
3 e^(3x) + C
e^(3x)/3 + C
e^(x)/3 + C
e^(3x) + C
Explanation - Integral of e^(kx) dx is (1/k) e^(kx) + C. Here k = 3, so ∫ e^(3x) dx = e^(3x)/3 + C.
Correct answer is: e^(3x)/3 + C
Q.9 Determine the derivative of f(x) = 1/x.
-1/x^2
1/x^2
-x
x^2
Explanation - Derivative of x^(-1) is -1 * x^(-2) = -1/x^2.
Correct answer is: -1/x^2
Q.10 Evaluate ∫ (3x^2 - 2x + 1) dx.
x^3 - x^2 + x + C
3x^3 - 2x^2 + x + C
x^3 - x^2 + 1 + C
x^3 - 2x^2 + x + C
Explanation - Integral of 3x^2 is x^3, integral of -2x is -x^2, integral of 1 is x, plus constant C.
Correct answer is: x^3 - x^2 + x + C
Q.11 Find the derivative of f(x) = tan(x).
sec(x)
sec^2(x)
cosec^2(x)
cot(x)
Explanation - Derivative of tan(x) with respect to x is sec^2(x).
Correct answer is: sec^2(x)
Q.12 If y = x * e^x, find dy/dx.
e^x
x e^x
e^x + x e^x
e^x - x e^x
Explanation - Use product rule: d(x*e^x)/dx = x*d(e^x)/dx + e^x*d(x)/dx = x*e^x + e^x = e^x + x e^x.
Correct answer is: e^x + x e^x
Q.13 Compute ∫ (1/x) dx.
ln|x| + C
1/x + C
x + C
e^x + C
Explanation - Integral of 1/x is ln|x| + C.
Correct answer is: ln|x| + C
Q.14 Evaluate the derivative of f(x) = sqrt(x).
1/(2 sqrt(x))
1/(sqrt(x))
1/(4 sqrt(x))
2 sqrt(x)
Explanation - Derivative of x^(1/2) is (1/2)x^(-1/2) = 1/(2 sqrt(x)).
Correct answer is: 1/(2 sqrt(x))
Q.15 Find ∫ (cos(x) - sin(x)) dx.
sin(x) + cos(x) + C
sin(x) - cos(x) + C
-sin(x) - cos(x) + C
-sin(x) + cos(x) + C
Explanation - Integral of cos(x) is sin(x), integral of -sin(x) is -cos(x), so ∫(cos(x)-sin(x)) dx = sin(x) - cos(x) + C.
Correct answer is: sin(x) - cos(x) + C
Q.16 If f(x) = x^4 - 2x^3 + x, find f''(x).
12x^2 - 6x + 1
4x^3 - 6x^2 + 1
12x^3 - 6x^2 + 1
12x^2 - 6x
Explanation - First derivative: f'(x) = 4x^3 - 6x^2 + 1. Second derivative: f''(x) = 12x^2 - 6x.
Correct answer is: 12x^2 - 6x + 1
Q.17 Evaluate ∫_0^π sin(x) dx.
0
1
2
-1
Explanation - Integral of sin(x) is -cos(x). Evaluate from 0 to π: -cos(π) + cos(0) = -(-1) + 1 = 2.
Correct answer is: 2
Q.18 Find the derivative of f(x) = x^2 ln(x).
2x ln(x) + x
2x ln(x) + x^2
x ln(x) + 2x
2x ln(x) - x
Explanation - Use product rule: d(x^2 ln(x))/dx = x^2*(1/x) + 2x*ln(x) = x + 2x ln(x) = 2x ln(x) + x.
Correct answer is: 2x ln(x) + x
Q.19 Compute ∫ (3/x) dx.
3 ln|x| + C
ln|3x| + C
ln|x| + C
x/3 + C
Explanation - Integral of k/x dx is k*ln|x| + C. Here k = 3, so ∫3/x dx = 3 ln|x| + C.
Correct answer is: 3 ln|x| + C
Q.20 If f(x) = cos(x)/x, find f'(x).
-sin(x)/x
-sin(x)/x - cos(x)/x^2
-sin(x)/x + cos(x)/x^2
sin(x)/x^2
Explanation - Use quotient rule: (u/v)' = (u'v - uv')/v^2. u = cos(x), v = x; u' = -sin(x), v' = 1. So f'(x) = (-sin(x)*x - cos(x)*1)/x^2 = (-x sin(x) - cos(x))/x^2 = -sin(x)/x - cos(x)/x^2.
Correct answer is: -sin(x)/x - cos(x)/x^2
Q.21 Evaluate ∫_1^2 (2/x) dx.
2 ln2
ln2
1
2
Explanation - Integral of 2/x dx is 2 ln|x|. Evaluate from 1 to 2: 2 ln2 - 2 ln1 = 2 ln2.
Correct answer is: 2 ln2
Q.22 Find the derivative of f(x) = e^(2x) sin(x).
2 e^(2x) sin(x) + e^(2x) cos(x)
e^(2x) cos(x) - 2 e^(2x) sin(x)
2 e^(2x) sin(x) - e^(2x) cos(x)
e^(2x) sin(x)
Explanation - Use product rule: u = e^(2x), v = sin(x). u' = 2 e^(2x), v' = cos(x). So f'(x) = u'v + uv' = 2 e^(2x) sin(x) + e^(2x) cos(x).
Correct answer is: 2 e^(2x) sin(x) + e^(2x) cos(x)
Q.23 Compute ∫ (x^3 + 2x) dx.
x^4 + x^2 + C
x^4/4 + x^2 + C
x^3 + x + C
x^4/4 + 2x + C
Explanation - Integral of x^3 is x^4/4, integral of 2x is x^2, add constant C: ∫(x^3 + 2x) dx = x^4/4 + x^2 + C.
Correct answer is: x^4/4 + x^2 + C
Q.24 Find the derivative of f(x) = ln(sin(x)).
cos(x)/sin(x)
-cos(x)/sin(x)
1/sin(x)
ln(cos(x))
Explanation - Derivative of ln(u) is u'/u. Here u = sin(x), u' = cos(x). So f'(x) = cos(x)/sin(x) = cot(x).
Correct answer is: cos(x)/sin(x)