AP Calculus: Differentiation - Composite and Implicit

This chapter introduces two powerful techniques for differentiating more complex functions. The first is the Chain Rule, which is the method for differentiating composite functions, or functions nested inside other functions, like sin(x^2). You will learn to differentiate the "outer" function first, leaving the inner function unchanged, and then multiply the result by the derivative of the "inner" function. The Chain Rule is used constantly throughout calculus, making it one of the most important rules to master.

The second technique is implicit differentiation. This method is used to find dy/dx for relations where y is not given as an explicit function of x, such as the equation of a circle, x² + y² = 25. The process involves differentiating both sides of the equation with respect to x, treating y as a function of x, and then algebraically solving for dy/dx. This chapter also typically covers the derivatives of inverse functions, including the inverse trigonometric functions (arcsin, arccos, etc.), and the calculation of higher-order derivatives. The most common difficulty is knowing when and how to apply the Chain Rule, especially multiple times in one problem. A solid command of all basic differentiation rules is essential before starting this chapter.

Formulas for AP Calculus: Differentiation - Composite and Implicit

The Chain Rule (Function Notation): $\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$ Differentiate the outer function, then multiply by the derivative of the inner function.
The Chain Rule (Leibniz Notation): $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$ Used when y is a function of u, and u is a function of x.
General Power Rule: $\frac{d}{dx}[u^n] = n u^{n-1} \frac{du}{dx}$ A common application of the Chain Rule where u is a function of x.
Implicit Differentiation of y: $\frac{d}{dx}[f(y)] = f'(y) \frac{dy}{dx}$ When differentiating a term with y with respect to x, multiply by dy/dx.
Derivative of an Inverse Function: $(f^{-1})'(a) = \frac{1}{f'(f^{-1}(a))}$ Gives the derivative of an inverse function at a point.
Derivative of arcsin(u): $\frac{d}{dx}[\arcsin(u)] = \frac{1}{\sqrt{1-u^2}} \frac{du}{dx}$ Here, u is a differentiable function of x.
Derivative of arctan(u): $\frac{d}{dx}[\arctan(u)] = \frac{1}{1+u^2} \frac{du}{dx}$ Here, u is a differentiable function of x.
Derivative of arcsec(u): $\frac{d}{dx}[\arcsec(u)] = \frac{1}{|u|\sqrt{u^2-1}} \frac{du}{dx}$ Here, u is a differentiable function of x.
Derivative of arccos(u): $\frac{d}{dx}[\arccos(u)] = -\frac{1}{\sqrt{1-u^2}} \frac{du}{dx}$ This is the negative of the derivative of arcsin(u).