Derivatives and Their Applications

What is the derivative of f(x) = 3x^4 + 2x^3 - x + 5?

• 6x^3 + 3x^2 - 1

• 12x^3 + 2x^2 + 5

• 9x^3 + 4x^2 - 1

• 12x^3 + 6x^2 - 1

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

• 18x^2 + 4x

• 9x^2 + 6x

• 12x^2 + 2

• 6x^2 + 3x

Calculate the derivative of g(x) = (x^2 + 1)/(x - 1) using the quotient rule.

• (2x)/(x - 1)

• (x^2 - 1)/(x + 1)^2

• (x^2 - 2x - 1)/(x - 1)^2

• (x^2 + 2x + 1)/(x - 1)^2

What is the derivative of h(x) = e^(2x)?

• e^(2x)

• e^(x)

• 2x e^(2x)

• 2e^(2x)

If f(x) = x^3 - 4x, what is f'(2)?

• 6

• 10

• 4

• 8

Using the product rule, find the derivative of f(x) = x^2 * e^x.

• f'(x) = x^2 * e^x

• f'(x) = 2e^x + x^2

• f'(x) = e^x(2x + x^2)

• f'(x) = e^x(2x^2 + x)

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

• g'(x) = 2/(x^2 + 1)

• g'(x) = 1/(x^2 + 1)

• g'(x) = x/(x^2 + 1)

• g'(x) = 2x/(x^2 + 1)

Calculate the derivative of f(x) = (x^2 + 3)(x^3 - 2) using the product rule.

• 4x^4 + 6x^2 - 2x

• 2x^4 + 7x^2 - 3x

• 5x^4 + 9x^2 - 4x

• 3x^4 + 8x^2 - 5x

What is the derivative of h(x) = 5e^(3x) - 2x^2?

• 5e^(3x) - 2x

• 5e^(3x) - 4x

• 15e^(3x) - 4x

• 15e^(2x) - 2x

Graphically, what does the derivative of a function represent at a given point?

• The area under the curve at that point.

• The slope of the tangent line at that point.

• The maximum value of the function at that point.

• The y-intercept of the function at that point.