Functions and Their Derivatives

• Differentiation Rule

• Power Rule

• Constant Rule

• Delta Method

It is defined as the derivative of the product of two differentiable functions is the first function times the derivative of the second function plus the second function times the derivative of the first function.

What Differentiation Rule is being defined??

What is the derivative of a constant?

• 0

• 1

• Any variable

• None of the mentioned choices.

If f(x) = sin³(2x), then f'(x) =

• 6sin²(2x)cos(2x)

• 6sin(2x)

• 3sin²(2x)cos(2x)

• 3cos(2x)

Find the derivative f(x) = xe^x

• f'(x) = e^x

• f'(x) = xe^x + xe^x

• f'(x) = e^x - xe^x

• f'(x) = e^x + xe^x

Find the derivative.
f(x) = -8x^-3 + 5x - e^x

• f'(x) = -24x^-4 + 5 - e^x

• f'(x) = 24x^-4 + 5 - e^x

• f'(x) = 24x^-2 + 5 - e^x

• f'(x) = 24x^-4 + 5 - e^x-1

Find the second derivative of
f(x) = x² + e^x  - cosx

• f"(x) = 2 + e^x + cosx

• f"(x) = 2x + e^x + cosx

• f"(x) = 2x + xe^x - cosx

• f"(x) = 2x + e^x + sinx

When we "take the derivative" of a function what are we finding?

• What's a derivative?

• The rate at which our struggles in Calculus are increasing.

• The slope of the secant line

• The slope of the tangent line