Chain Rule Review

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^6 + 4)^5.

Back

f '(x) = 30x^5(x^6 + 4)^4.

Front

Differentiate f(x) = (7/5) sin^2(2x).

Back

f '(x) = (28 sin(2x) cos(2x))/5.

Front

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

Back

f '(x) = 24x^7(5 + 8x)^2 + 7x^6(5 + 8x)^3.

Front

Differentiate y = tan(3x^2 + 2).

Back

y' = 6x sec^2(3x^2 + 2).

Front

What is the derivative of sin^2(x)?

Back

d/dx (sin^2(x)) = 2(sin(x))(cos(x)).

Front

What is the power rule for differentiation?

Back

The power rule states that if f(x) = x^n, then f'(x) = n*x^(n-1).

Front

What is the product rule for differentiation?

Back

The product rule states that if u(x) and v(x) are functions, then the derivative of their product is u'v + uv'.

Front

What is the quotient rule for differentiation?

Back

The quotient rule states that if u(x) and v(x) are functions, then the derivative of their quotient is (u'v - uv')/v^2.

Front

How do you differentiate composite functions?

Back

Use the Chain Rule: if y = f(g(x)), then y' = f'(g(x)) * g'(x).