12-20 All Derivative Rules Practice

Front

What is the Power Rule for differentiation?

Back

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

Front

Find the derivative of \( y = (3x + 2)^5 \).

Back

Using the chain rule, \( y' = 15(3x + 2)^4 \).

Front

What is the Chain Rule in differentiation?

Back

The Chain Rule states that if a function \( y = f(g(x)) \), then \( y' = f'(g(x)) \cdot g'(x) \).

Front

Find \( f'(x) \) if \( f(x) = e^{2x} \).

Back

Using the chain rule, \( f'(x) = 2e^{2x} \).

Front

What is the Product Rule for differentiation?

Back

The Product Rule states that if \( y = u(x)v(x) \), then \( y' = u'v + uv' \).

Front

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

Back

Using the product rule, \( y' = 2x \sin(x) + x^2 \cos(x) \).

Front

What is the Quotient Rule for differentiation?

Back

The Quotient Rule states that if \( y = \frac{u(x)}{v(x)} \), then \( y' = \frac{u'v - uv'}{v^2} \).

Front

Find the derivative of \( y = \frac{1}{x^2} \).

Back

Using the power rule, \( y' = -2x^{-3} = -\frac{2}{x^3} \).

Front

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

Back

The derivative of \( \cos(x) \) is \( -\sin(x) \).

Front

Find \( f'(x) \) if \( f(x) = \sin(3x) \).

Back

Using the chain rule, \( f'(x) = 3\cos(3x) \).