Q.1 What is the derivative of x^2 with respect to x?
x
2x
x^2
2
Explanation - Using the power rule, d/dx(x^n) = n*x^(n-1). Here n=2, so derivative is 2x.
Correct answer is: 2x
Q.2 What is the derivative of sin(x)?
cos(x)
-cos(x)
sin(x)
-sin(x)
Explanation - The derivative of sin(x) with respect to x is cos(x).
Correct answer is: cos(x)
Q.3 What is d/dx(e^x)?
1
e^x
ln(x)
x^e
Explanation - The exponential function e^x is its own derivative.
Correct answer is: e^x
Q.4 Find the derivative of ln(x).
1/x
ln(x)
x
e^x
Explanation - The derivative of ln(x) is 1/x for x>0.
Correct answer is: 1/x
Q.5 Differentiate cos(x).
sin(x)
-sin(x)
cos(x)
-cos(x)
Explanation - The derivative of cos(x) is -sin(x).
Correct answer is: -sin(x)
Q.6 What is the derivative of x^3?
3x^2
2x
x^2
3x
Explanation - Using power rule, derivative of x^3 is 3x^2.
Correct answer is: 3x^2
Q.7 If y = 5x, find dy/dx.
5
x
0
1
Explanation - The derivative of ax is a. Here a=5, so dy/dx = 5.
Correct answer is: 5
Q.8 Differentiate tan(x).
sec^2(x)
cos^2(x)
-sec^2(x)
1 + tan^2(x)
Explanation - The derivative of tan(x) is sec^2(x).
Correct answer is: sec^2(x)
Q.9 Find the derivative of 7.
0
7
1
x
Explanation - The derivative of any constant is zero.
Correct answer is: 0
Q.10 Find d/dx of x^(-1).
1/x^2
-1/x^2
x^-2
-x^-2
Explanation - Using power rule, derivative of x^(-1) = -1*x^(-2) = -1/x^2.
Correct answer is: -1/x^2
Q.11 What is derivative of sqrt(x)?
1/(2√x)
√x
2√x
x^(3/2)
Explanation - sqrt(x) = x^(1/2). Using power rule, derivative = (1/2)x^(-1/2) = 1/(2√x).
Correct answer is: 1/(2√x)
Q.12 Find the derivative of log10(x).
1/(x ln(10))
1/x
ln(x)
10/x
Explanation - Derivative of log base a of x is 1/(x ln(a)). For log10(x), it is 1/(x ln(10)).
Correct answer is: 1/(x ln(10))
Q.13 If y = 3x^2 + 2x + 1, what is dy/dx?
3x + 2
6x + 2
2x + 2
6x
Explanation - Derivative term by term: 3x^2 → 6x, 2x → 2, constant → 0. So result is 6x + 2.
Correct answer is: 6x + 2
Q.14 What is derivative of sin^2(x)?
2sin(x)
2sin(x)cos(x)
cos^2(x)
1 - cos^2(x)
Explanation - Using chain rule: d/dx(sin^2(x)) = 2sin(x)*cos(x).
Correct answer is: 2sin(x)cos(x)
Q.15 Differentiate e^(2x).
2e^(2x)
e^(2x)
ln(2)e^(2x)
x e^(2x)
Explanation - d/dx(e^(kx)) = k e^(kx). Here k=2, so derivative = 2e^(2x).
Correct answer is: 2e^(2x)
Q.16 If f(x) = 1/x, then f'(x) = ?
-1/x^2
1/x
ln(x)
0
Explanation - Derivative of 1/x is -1/x^2 using power rule.
Correct answer is: -1/x^2
Q.17 Find derivative of sec(x).
sec(x)tan(x)
-sec(x)tan(x)
tan(x)
1/cos^2(x)
Explanation - Derivative of sec(x) is sec(x)tan(x).
Correct answer is: sec(x)tan(x)
Q.18 Differentiate cot(x).
-csc^2(x)
csc^2(x)
-cos(x)/sin(x)
tan(x)
Explanation - Derivative of cot(x) is -csc^2(x).
Correct answer is: -csc^2(x)
Q.19 What is d/dx of arcsin(x)?
1/√(1-x^2)
-1/√(1-x^2)
√(1-x^2)
1/(1+x^2)
Explanation - Derivative of arcsin(x) is 1/√(1-x^2).
Correct answer is: 1/√(1-x^2)
Q.20 Find derivative of arctan(x).
1/(1+x^2)
1/(1-x^2)
arctan'(x)
1/√(1-x^2)
Explanation - Derivative of arctan(x) is 1/(1+x^2).
Correct answer is: 1/(1+x^2)
Q.21 Differentiate x^4 + 2x^3 - x + 7.
4x^3 + 6x^2 -1
4x^3 + 2x^2 -1
x^3 + 3x^2 -1
4x^3 - 1
Explanation - Derivative term by term: x^4→4x^3, 2x^3→6x^2, -x→-1, constant→0.
Correct answer is: 4x^3 + 6x^2 -1
Q.22 If y = ln(√x), then dy/dx = ?
1/(2x)
1/x
2/x
√x/x
Explanation - ln(√x) = ln(x^(1/2)) = (1/2)ln(x). Derivative = 1/(2x).
Correct answer is: 1/(2x)
Q.23 Find derivative of x^5 - 3x^2 + 4.
5x^4 - 6x
x^4 - 6x
5x^3 - 6
5x^4 - 3x
Explanation - Using power rule, derivative is 5x^4 - 6x.
Correct answer is: 5x^4 - 6x
Q.24 What is d/dx of e^(x^2)?
2x e^(x^2)
e^(x^2)
x e^(x^2)
2 e^(x)
Explanation - Using chain rule, derivative = e^(x^2) * d/dx(x^2) = 2x e^(x^2).
Correct answer is: 2x e^(x^2)
Q.25 Find derivative of sin(3x).
3cos(3x)
cos(3x)
sin(3x)
-3sin(3x)
Explanation - Using chain rule, derivative = cos(3x)*3 = 3cos(3x).
Correct answer is: 3cos(3x)