Calculus Derivatives and Rules

This video tutorial covers the product, quotient, and chain rules for derivatives, providing detailed explanations and examples for each. It also demonstrates how to apply these rules using data tables, helping students understand the practical application of these concepts in calculus.

What is the product rule used for in calculus?

• Finding the integral of a function

• Finding the limit of a function

• Finding the derivative of a product of two functions

• Finding the derivative of a sum of functions

In the product rule, if f(x) = x^2 and g(x) = e^x, what is f'(x)g(x)?

• 2x * e^x

• x^2 * e^x

• 2x * x^2

• e^x * e^x

What is the quotient rule formula for finding the derivative of f(x)/g(x)?

• (f(x)g'(x) - g(x)f'(x))/f(x)^2

• (g(x)f'(x) + f(x)g'(x))/f(x)^2

• (g(x)f'(x) - f(x)g'(x))/g(x)^2

• (f'(x)g(x) + f(x)g'(x))/g(x)^2

What is the derivative of the function y = (3x - 1)/(2x + 1) using the quotient rule?

• (6x + 2)/(2x + 1)^2

• (5)/(2x + 1)^2

• (3x - 1)/(2x + 1)^2

• (6x - 2)/(2x + 1)^2

What is a composite function in the context of the chain rule?

• A function inside another function

• A function that is the product of two functions

• A function that is the sum of two functions

• A function that is the quotient of two functions

Using the chain rule, what is the derivative of y = (x^3 - 5x)^4?

• 4(x^3 - 5x)^3 * (2x^2 - 5)

• 4(x^3 - 5x)^3 * (3x^2 - 5)

• 3(x^3 - 5x)^4 * (3x^2 - 5)

• 4(x^3 - 5x)^4 * (3x^2 - 5)

What is the derivative of y = sin(x^2) using the chain rule?

• sin(x^2) * 2x

• 2x * sin(x^2)

• 2x * cos(x^2)

• cos(x^2)

How do you find h'(2) if h(x) = f(x) * g(x) using a data table?

• f'(2) * g(2) + f(2) * g'(2)

• f(2) * g(2) + f'(2) * g'(2)

• f'(2) * g'(2) + f(2) * g(2)

• f'(2) * g(2) - f(2) * g'(2)

If h(x) = f(g(x)), what is the formula for h'(x) using the chain rule?

• f(g(x)) + g'(x)

• f'(g(x)) + g'(x)

• f(g(x)) * g'(x)

• f'(g(x)) * g'(x)

What is the value of h'(1) if h(x) = g(g(x)) using a data table?

• 8

• -8

• 16

• -16