Mastering Differentiation Concepts

What is the derivative of f(x) = 3x^2 + 5?

• 6x

• 3x

• 5x

• 6x^2

Using the product rule, find the derivative of f(x) = (2x)(x^3 + 1).

• 8x^3 + 2

• 2x^4 + 2

• 8x^2 + 3

• 6x^3 + 2

Differentiate the function f(x) = x^4 - 4x^3 + 6x - 1.

• 4x^3 - 10x^2 + 7

• 4x^3 - 12x^2 + 6

• 3x^3 - 12x^2 + 5

• 4x^3 - 8x^2 + 6

Apply the quotient rule to find the derivative of f(x) = (x^2 + 1)/(x - 1).

• (x^2 - 2x - 1)/(x - 1)^2

• (x^2 + 2x + 1)/(x - 1)^2

• (2x)/(x - 1)

• (x^2 - 1)/(x - 1)^2

What is the chain rule and how is it applied in differentiation?

• The chain rule is a method for finding the area under a curve by averaging the values of the function.

• The chain rule allows us to differentiate composite functions by multiplying the derivatives of the outer and inner functions.

• The chain rule is used to integrate functions by adding the derivatives of the outer and inner functions.

• The chain rule states that the derivative of a function is always zero regardless of its composition.

Find the second derivative of f(x) = sin(x).

• sin(2x)

• cos(x)

• -cos(x)

• -sin(x)

Differentiate the function f(x) = e^(2x) using the chain rule.

• 2e^(2x)

• e^(x)

• 4e^(2x)

• e^(2x)

What is the derivative of f(x) = ln(x^2 + 1)?

• 2/(x^2 + 1)

• x/(x^2 + 1)

• 1/(x^2 + 1)

• 2x/(x^2 + 1)

Using the product rule, differentiate f(x) = (x^2)(sin(x)).

• f'(x) = 2x sin(x) + x^2 cos(x)

• f'(x) = 2x^2 cos(x) + sin(x)

• f'(x) = x^2 sin(x) + 2x cos(x)

• f'(x) = 2x cos(x) + x^2 sin(x)

Find the first and second derivatives of f(x) = 5x^3 - 3x + 2.

• f'(x) = 10x^2 - 3, f''(x) = 20x

• f'(x) = 15x^2 - 6, f''(x) = 30

• f'(x) = 15x^2 + 3, f''(x) = 30x^2

• f'(x) = 15x^2 - 3, f''(x) = 30x