AP Calc unit 3/4 review

Front

What is the Quotient Rule in calculus?

Back

The Quotient Rule states that if you have a function that is the quotient of two functions, say \( f(x) = \frac{g(x)}{h(x)} \), then the derivative is given by \( f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2} \).

Front

When do you use the Chain Rule?

Back

The Chain Rule is used when differentiating a function that is composed of other functions, i.e., when there is a function within a function.

Front

Find \( \frac{dy}{dx} \) if \( y = \ln(2x) \).

Back

\( \frac{dy}{dx} = \frac{1}{x} \)

Front

What is the product of the derivatives of two functions?

Back

If \( f(x) \) and \( g(x) \) are two functions, then the derivative of their product is given by the Product Rule: \( (fg)' = f'g + fg' \).

Front

Find \( \frac{dy}{dx} \) if \( y = (2x^2 + 4)^3 \).

Back

\( \frac{dy}{dx} = 12x(2x^2 + 4)^2 \)

Front

What is the definition of a derivative?

Back

The derivative of a function at a point is the limit of the average rate of change of the function as the interval approaches zero, representing the slope of the tangent line at that point.

Front

What is the Chain Rule formula?

Back

If \( y = f(g(x)) \), then \( \frac{dy}{dx} = f'(g(x))g'(x) \).

Front

What is the significance of the natural logarithm in calculus?

Back

The natural logarithm, denoted as \( \ln(x) \), is the inverse of the exponential function and has unique properties that simplify differentiation and integration.

Front

What is implicit differentiation?

Back

Implicit differentiation is a technique used to find the derivative of a function defined implicitly by an equation, rather than explicitly as \( y = f(x) \).

Front

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

Back

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