CC Chain Rule

Front

What is the Chain Rule in calculus?

Back

The Chain Rule is a formula for computing the derivative of the composition of two or more functions. It states that if you have a function y = f(g(x)), then the derivative y' = f'(g(x)) * g'(x).

Front

Differentiate: f(x) = x^7 (5 + 8x)^3

Back

f '(x) = x^6 (5 + 8x)^2 (35 + 80x) using the product and chain rules.

Front

Differentiate: f(x) = (x^6 + 4)^5

Back

f '(x) = 30x^5 (x^6 + 4)^4 using the chain rule.

Front

What is the derivative of a constant?

Back

The derivative of a constant is zero.

Front

How do you find the derivative of a product of two functions?

Back

Use the Product Rule: If u and v are functions, then (uv)' = u'v + uv'.

Front

How do you find the derivative of a quotient of two functions?

Back

Use the Quotient Rule: If u and v are functions, then (u/v)' = (u'v - uv')/v^2.

Front

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

Back

f '(x) = nx^(n-1), where n is a constant.

Front

What is the significance of the derivative at a point?

Back

The derivative at a point gives the slope of the tangent line to the graph of the function at that point.

Front

How do you find the equation of a tangent line at a point?

Back

Use the point-slope form: y - y1 = m(x - x1), where m is the derivative at x1.

Front

Differentiate: f(x) = (5x^5 + 5)^2

Back

f '(x) = 500x^4 (5x^5 + 5) using the chain rule.