Understanding Derivatives

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

• x^3

• 2x

• 2x^2

• x

Find the derivative of f(x) = 3x^3 - 5x + 2.

• 9x^2 - 5

• 3x^2 - 5

• 6x^2 - 5

• 9x^2 + 2

What is the derivative of f(x) = sin(x)?

• cos(x)

• sin^2(x)

• cosh(x)

• tan(x)

Calculate the derivative of f(x) = e^x.

• e^2x

• x^e

• e^x

• ln(x)

What is the derivative of f(x) = ln(x)?

• ln(x^2)

• x

• 1/x^2

• 1/x

Find the derivative of f(x) = cos(x).

• -cos(x)

• sin(x)

• -sin(x)

• tan(x)

What is the product rule for derivatives?

• (u*v)' = u'v' + uv

• (u*v)' = u'v' - uv'

• (u*v)' = u' + v'

• (u*v)' = u'v + uv'

Explain the quotient rule for derivatives.

• (u/v)' = (u'v + uv') / v

• The quotient rule for derivatives is (u/v)' = (u'v - uv') / v^2.

• (u/v)' = (u'v + uv') / v^2

• (u/v)' = (u'v - uv') / v

What is the chain rule in differentiation?

• The chain rule applies only to linear functions.

• The chain rule is used to find the integral of a function.

• The chain rule states that dy/dx = f'(x) * g'(x).

• The chain rule in differentiation states that if y = f(g(x)), then dy/dx = f'(g(x)) * g'(x).

Find the second derivative of f(x) = x^4.

• 12x^2

• x^2

• 8x^3

• 4x^3