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)