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(z) = 1/(3z^2) + 4z^3 using the Chain Rule.

Back

f'(z) = -2/(3z^3) + 12z^2.

Front

Differentiate y = (2x + 1)^10 using the Chain Rule.

Back

y' = 20(2x + 1)^9.

Front

Differentiate y = e^(5x) + 1 using the Chain Rule.

Back

y' = 5e^(5x).

Front

Differentiate f(x) = (x^6 + 4)^5 using the Chain Rule.

Back

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

Front

Differentiate f(x) = (x^3 - 2x)^2 using the Chain Rule.

Back

f'(x) = 6x^5 - 16x^3 + 8x.

Front

What is the derivative of a constant function?

Back

The derivative of a constant function is zero.

Front

How do you apply the Chain Rule to find the derivative of composite functions?

Back

Identify the outer function and inner function, differentiate the outer function, then multiply by the derivative of the inner function.

Front

What is the derivative of y = sin(g(x))?

Back

y' = cos(g(x)) * g'(x) (using the Chain Rule).

Front

What is the derivative of y = ln(g(x))?

Back

y' = (1/g(x)) * g'(x) (using the Chain Rule).