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.