### Chain rule differentiation integration mcq

What does the chain rule allow us to do?

A) Differentiate composite functions

B) Integrate trigonometric functions

C) Find indefinite integrals

D) Solve linear equations

Answer: A) Differentiate composite functions

The chain rule is a rule in:

A) Algebra

B) Geometry

C) Calculus

D) Statistics

Answer: C) Calculus

Which of the following functions is an example of a composite function ?

A) f(x) = x^2

B ) g(x) = 2x + 3

C) h(x) = sin(x)

D) k(x) = ln(x)

Answer: C) h(x) = sin(x)

The chain rule is used to differentiate functions of the form:

A) f(g(x))

B) f(x)g(x)

C) f (x) + g(x)

D) f (x) / g(x)

Answer: A) f(g(x))

What is the derivative of f(g(x)) with respect to x, using the chain rule?

A) f'(g(x)) * g'(x)

B) f'(x) * g ‘(x)

C) f(g(x))

D) f(x) + g (x)

Answer: A) f'(g(x)) * g'(x)

If f(x) = x^3 and g(x) = 2x + 1 , what is the derivative of f (g(x)) with respect to x?

A) 3x^2 + 2

B) 6x^2 + 1

C) 6x + 3

D) 3x^2 + 1

Answer: B) 6x^2 + 1

The chain rule is used to differentiate which of the following functions?

A) Constant functions

B) Linear functions

C) Exponential functions

D) Quadratic functions

Answer: C) Exponential functions

If f(x) = sin(2x) and g(x) = x^2, what is the derivative of f(g (x) ) with respect to x ?

A) 2cos(2x)

B) 2xcos(2x)

C) cos(2x)

D) x ^2cos (2x)

Answer: B) 2xcos(2x)

The chain rule is a consequence of which rule in calculus?

A) Product rule

B) Quotient rule

C) Power rule

D) Sum rule

Answer: A) Product rule

If f(x) = ln(2x) and g(x) = x^3, what is the derivative of f (g (x)) with respect to x?

A) (3x^2) / (x^3)

B) (3ln(2x))/(2x)

C) (3ln(2x))/(x^2)

D) (2x) / (3ln (2x))

Answer: C) (3ln(2x))/(x^2)

The chain rule can be used to differentiate which of the following functions?

A) Polynomial functions

B) Logarithmic functions

C) Trigonometric functions

D) Rational functions

Answer: B) Logarithmic functions

If f(x) = e^( 2x) and g(x) = cos(x) , what is the derivative of f(g( x )) with respect to x?

A) 2e^(2x) sin(x)

B) 2e^(2x) cos(x)

C) e^(2x )sin(x)

D) e^(2x) cos(x)

Answer: B) 2e^(2x)cos(x)

The chain rule is used to differentiate which of the following functions?

A) Absolute value functions

B) Square root functions

C) Step functions

D) Trigonometric functions

Answer: D) Trigonometric functions

If f(x) = sqrt( x) and g(x) = 2x + 1 , what is the derivative of f (g(x) ) with respect to x?

A) (2x + 1) / (2sqrt(x))

B) (2x + 1) / sqrt(x)

C) 2sqrt(x) + 1

D) sqrt(2x + 1)

Answer: A) (2x + 1) / (2sqrt(x))

The chain rule can be used to differentiate functions that involve which mathematical operation?

A) Addition

B) Subtraction

C) Multiplication

D) Division

Answer: C) Multiplication

If f(x) = e ^(3x) and g(x) = ln (x) , what is the derivative of f(g(x)) with respect to x?

A) 3e^(3x) / x

B) 3e^(3x) / ln(x)

C) e^(3x) / ln(x)

D) e^(3x) / x

Answer: D) e^(3x) / x

The chain rule is applicable to which type of functions?

A) Continuous functions

B) Discontinuous functions

C) Linear functions

D) Polynomial functions

Answer: A) Continuous functions

If f(x) = 2x^3 and g(x) = e^x, what is the derivative of f(g(x)) with respect to x?

A) 3x^2 e^x

B) 6x^2 e^x

C) 6x^2 e^(2x)

D) 3x^2 e^(2x)

Answer: C) 6x^2e^(2x)

The chain rule is used to differentiate which of the following expressions?

A) x^2 + 1

B) 2x^3 – 5x^2

C) 3sin(x) + 2cos(x)

D) 4x^2 + 3x + 1

Answer: C) 3sin(x) + 2cos(x)

If f(x) = 3x^2 and g (x) = ln(x^2) , what is the derivative of f(g(x)) with respect to x

A) 12x / ln(x^2)

B) 6x / ln(x)

C) 6x / ln(x^2)

D) 6x^2 / ln(x)

Answer: C) 6x / ln(x^2)

The chain rule can be used to differentiate which of the following functions?

A) Exponential functions

B) Logarithmic functions

C) Trigonometric functions

D) All of the above

Answer: D) All of the above

If f(x) = 2 ^ x and g(x) = sin(x), what is the derivative of f (g(x)) with respect to x?

A) 2^x * cos(x)

B) 2^x * cos(x)

C) ln(2) * cos(x)

D) ln(2) * sin(x)

Answer: C) ln(2) * cos(x)

The chain rule is a fundamental concept in which branch of mathematics?

A) Algebra

B) Geometry

C) Calculus

D) Trigonometry

Answer: C) Calculus

If f(x) = tan(x) and g( x) = e^x, what is the derivative of f (g(x) ) with respect to x ?

A) sec^2(x) * e^x

B) sec^2(x) * e^x

C) tan(x) * e^x

D) tan(x) * e^x

Answer: B) sec^2( x) * e^x

The chain rule can be used to differentiate which of the following expressions?

A) x^2 + x + 1

B) 2x^3 – 3x^2 + 4x – 5

C) 3sin(x) + 2cos(x) + 1

D) 4x^2 + 3x + 2

Answer: C) 3sin(x) + 2cos(x) + 1

If f(x) = 3x ^2 and g(x) = e ^ (2x), what is the derivative of f(g(x) ), with respect to x?

A) 12x * e^(2x)

B) 6x * e^(2x)

C) 6x * e^(2x)

D) 6x^2 * e^(2x)

Answer: C) 6x * e^(2x)

The chain rule can be used to differentiate which of the following functions?

A) Linear functions

B) Quadratic functions

C) Exponential functions

D) Absolute value functions

Answer: C) Exponential functions

If f(x) = sqrt(x) and g(x) = cos(2x) , what is the derivative of f(g(x)) with respect to x?

A) -sin(2x) / (2sqrt(x))

B) -sin(2x) / sqrt(x)

C) -2sin(2x) / (2sqrt(x))

D) -2sin(2x) / sqrt(x)

Answer: D) -2sin(2x) / sqrt(x)

The chain rule is used to differentiate which of the following functions?

A) Polynomial functions

B) Logarithmic functions

C) Trigonometric functions

D) Rational functions

Answer: D) Rational functions

If f(x) = ln(2x) and g(x) = x^3, what is the derivative of f(g(x)) with respect to x?

A) 3x^2 / (2x)

B) 3x^2 / (2x^3)

C) 3x^2 / (2x^2)

D) 3x^2 / (2ln(2x))

Answer: A) 3x^2 / (2x)