Chain Rule Practice

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 two functions f(x) and g(x), then the derivative of their composition is given by: \( (f(g(x)))' = f'(g(x)) \cdot g'(x) \).

Front

Find the derivative of \( y = \sin(2x^2) \).

Back

\( \frac{dy}{dx} = 4x \cos(2x^2) \) (using the Chain Rule).

Front

What is the derivative of \( f(x) = \cos(5x^4) \)?

Back

\( f'(x) = -20x^3 \sin(5x^4) \) (using the Chain Rule).

Front

If \( y = \ln(x - 2) \), what is \( y' \)?

Back

\( y' = \frac{1}{x - 2} \) (using the derivative of the natural logarithm).

Front

Find the derivative of \( h(x) = e^{4x} \).

Back

\( h'(x) = 4e^{4x} \) (using the derivative of the exponential function).

Front

What is the derivative of \( f(x) = (3x^2 + 2x)^4 \)?

Back

\( f'(x) = 4(3x^2 + 2x)^3(6x + 2) \) (using the Chain Rule and the Power Rule).

Front

What is the Power Rule in calculus?

Back

The Power Rule states that if \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \).

Front

What is the derivative of \( \sin(x) \)?

Back

\( \frac{d}{dx} \sin(x) = \cos(x) \).

Front

What is the derivative of \( \cos(x) \)?

Back

\( \frac{d}{dx} \cos(x) = -\sin(x) \).

Front

What is the derivative of \( e^x \)?

Back

\( \frac{d}{dx} e^x = e^x \).