Derivatives of Combined Functions

Overview

This section covers the rules for finding derivatives of combinations of functions, specifically:
* Sum/Difference Rule: $f(x) \pm g(x)$
* Product Rule: $f(x) \times g(x)$
* Quotient Rule: $\frac{f(x)}{g(x)}$
* Chain Rule (Composition of Functions): $(f \circ g)(x) = f(g(x))$

Where $f$ and $g$ are polynomial, exponential, circular (trigonometric), logarithmic, or power functions (or simple combinations thereof).

1. Sum and Difference Rule

Key Idea

The derivative of a sum (or difference) of functions is the sum (or difference) of their derivatives.

Rule

If $f(x) = u(x) + v(x)$, then $f’(x) = u’(x) + v’(x)$.
If $f(x) = u(x) - v(x)$, then $f’(x) = u’(x) - v’(x)$.

Examples

• If $f(x) = x^2 + 3x$, then $f’(x) = 2x + 3$.
• If $f(x) = \sin(x) - e^x$, then $f’(x) = \cos(x) - e^x$.

2. Product Rule

Key Idea

The derivative of a product of two functions requires a specific formula.

Rule

If $f(x) = u(x)v(x)$, then $f’(x) = u’(x)v(x) + u(x)v’(x)$.

Mnemonic

“First times derivative of the second, plus second times derivative of the first.”

Examples

• If $f(x) = x^2\sin(x)$, then $f’(x) = 2x\sin(x) + x^2\cos(x)$.
• If $f(x) = e^x\ln(x)$, then $f’(x) = e^x\ln(x) + e^x(\frac{1}{x})$.

3. Quotient Rule

Key Idea

The derivative of a quotient of two functions also requires a specific formula.

Rule

If $f(x) = \frac{u(x)}{v(x)}$, then $f’(x) = \frac{u’(x)v(x) - u(x)v’(x)}{[v(x)]^2}$.

Mnemonic

“Low d’High minus High d’Low, over Low squared.”

Examples

• If $f(x) = \frac{x^3}{x+1}$, then $f’(x) = \frac{3x^2(x+1) - x^3(1)}{(x+1)^2} = \frac{2x^3 + 3x^2}{(x+1)^2}$.
• If $f(x) = \frac{\sin(x)}{x}$, then $f’(x) = \frac{\cos(x) \cdot x - \sin(x) \cdot 1}{x^2} = \frac{x\cos(x) - \sin(x)}{x^2}$.

4. Chain Rule

Key Idea

The chain rule is used to find the derivative of a composite function.

Rule

If $f(x) = u(v(x))$, then $f’(x) = u’(v(x)) \cdot v’(x)$.

Mnemonic

“Derivative of the outside, evaluated at the inside, times the derivative of the inside.”

Examples

• If $f(x) = (x^2 + 1)^3$, then $f’(x) = 3(x^2 + 1)^2 \cdot 2x = 6x(x^2 + 1)^2$.
• If $f(x) = \sin(e^x)$, then $f’(x) = \cos(e^x) \cdot e^x = e^x\cos(e^x)$.
• If $f(x) = e^{\sin(x)}$, then $f’(x) = e^{\sin(x)} \cdot \cos(x) = \cos(x)e^{\sin(x)}$.

Summary Table

Common Derivatives

It’s helpful to know the derivatives of common functions:

• $\frac{d}{dx}(x^n) = nx^{n-1}$
• $\frac{d}{dx}(\sin(x)) = \cos(x)$
• $\frac{d}{dx}(\cos(x)) = -\sin(x)$
• $\frac{d}{dx}(e^x) = e^x$
• $\frac{d}{dx}(\ln(x)) = \frac{1}{x}$

Applications

These differentiation rules are fundamental to solving a wide range of problems in calculus, including finding stationary points, rates of change, and optimization problems.