Chain Rule Differentiation

Front

What is the Chain Rule in differentiation?

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

Find y' for y = (3x + 2)^5.

Back

y' = 15(3x + 2)^4 * 3 = 45(3x + 2)^4.

Front

What is the derivative of f(x) = (x^2 + 1)^3?

Back

f'(x) = 3(x^2 + 1)^2 * 2x = 6x(x^2 + 1)^2.

Front

When do you apply the Chain Rule?

Back

You apply the Chain Rule when differentiating a function that is composed of another function, i.e., when there is a function within a function.

Front

Find the derivative of h(x) = 5(x^3 - 4)^2.

Back

h'(x) = 10(x^3 - 4)(3x^2) = 30x^2(x^3 - 4).

Front

What is the Power Rule?

Back

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

Front

Find the derivative of f(x) = (2x^2 + 3x)^4.

Back

f'(x) = 4(2x^2 + 3x)^3 * (4x + 3).

Front

What is the Product Rule?

Back

The Product Rule is used to find the derivative of the product of two functions. If u(x) and v(x) are functions, then (uv)' = u'v + uv'.

Front

Find y' for y = (x^2 + 1)(x^3 - 2).

Back

y' = (2x)(x^3 - 2) + (x^2 + 1)(3x^2) = 2x(x^3 - 2) + 3x^2(x^2 + 1).

Front

What is the Quotient Rule?

Back

The Quotient Rule is used to find the derivative of the quotient of two functions. If u(x) and v(x) are functions, then (u/v)' = (u'v - uv')/v^2.