Test Review (Power Rule, AROC, Tangent/Normal Line)

Front

What is the Power Rule in differentiation?

Back

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

Front

Differentiate the function: \( y = 4x^3 + 3x + 1 \)

Back

\( \frac{dy}{dx} = 12x^2 + 3 \) (using the Power Rule).

Front

Differentiate the function: \( f(x) = x^3 + x^2 + 3 \)

Back

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

Front

Differentiate the function: \( y = \frac{2x^2 + x - 3}{x} \)

Back

First simplify to \( y = 2x + 1 - 3x^{-1} \), then \( y' = 2 + 3x^{-2} \).

Front

What does the General Power Rule state?

Back

The General Power Rule states that \( \frac{d}{dx}[x^n] = nx^{n-1} \).

Front

Differentiate the function: \( y = 4x^2 + 3x + 6x^{-2} \)

Back

\( y' = 8x + 3 - 12x^{-3} \) (using the Power Rule).

Front

What is the derivative of a constant?

Back

The derivative of a constant is 0.

Front

What is the derivative of \( x^n \) where n is a constant?

Back

The derivative is \( nx^{n-1} \) (Power Rule).

Front

How do you find the derivative of a quotient?

Back

Use the Quotient Rule: if \( y = \frac{u}{v} \), then \( y' = \frac{u'v - uv'}{v^2} \).

Front

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

Back

The derivative of \( e^x \) is \( e^x \).