AP Calculus AB/BC - Unit 3: Differentiation

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Derivative of a Composite Function:

If $f$ and $g$ are differentiable, then the derivative of $f \circ g$ is given by:

$(f \circ g)' = \frac{d}{dx} \left[ f(g(x)) \right] = f'(g(x)) g'(x)$

This follows directly from the chain rule.

Derivative of an Inverse Function:

If $f$ has an inverse function $g$ , then:

$g'(x) = \frac{1}{f'(g(x))}$

Derivatives of inverse functions are **reciprocal slopes**.

Implicit Differentiation:

In implicit differentiation, you will have a $\frac{dy}{dx}$ for each $y$ in the original function or equation. The steps are:

Differentiate both sides of the equation with respect to $x$ .
Apply the chain rule wherever $y$ appears.
Solve for $\frac{dy}{dx}$ .

If you are taking the second derivative $\frac{d^2 y}{dx^2}$ , you will often substitute the expression you found for the first derivative somewhere in the process.