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Rules of Differentiation

Simple Idea

  • Calculating derivatives using the limit definition (\(\lim [f(x+h)-f(x)]/h\)) can be tedious. Thankfully, there are shortcut rules for finding derivatives of common types of functions quickly.

Formal Definition

  • These rules are derived from the limit definition of the derivative but allow for mechanical calculation once memorized. We assume \(f(x)\) and \(g(x)\) are differentiable functions and \(c\) is a constant.

Key Rules

1. Constant Rule

  • The derivative of a constant function is zero. $$ \frac{d}{dx}© = 0 $$
  • Intuition: A constant function \(f(x)=c\) has a horizontal graph, so its slope (derivative) is always 0.

2. Power Rule

  • The derivative of \(x\) raised to a power \(n\) (where \(n\) is any real number) is \(n\) times \(x\) raised to the power \(n-1\). $$ \frac{d}{dx}(x^n) = n x^{n-1} $$
  • Examples:
    • \(\frac{d}{dx}(x^3) = 3x^2\)
    • \(\frac{d}{dx}(x) = \frac{d}{dx}(x^1) = 1x^0 = 1\)
    • \(\frac{d}{dx}(\sqrt{x}) = \frac{d}{dx}(x^{1/2}) = \frac{1}{2}x^{-1/2} = \frac{1}{2\sqrt{x}}\)
    • \(\frac{d}{dx}(\frac{1}{x}) = \frac{d}{dx}(x^{-1}) = -1x^{-2} = -\frac{1}{x^2}\)

3. Constant Multiple Rule

  • The derivative of a constant times a function is the constant times the derivative of the function. $$ \frac{d}{dx}[c \cdot f(x)] = c \cdot \frac{d}{dx}[f(x)] = c \cdot f'(x) $$
  • Intuition: Scaling a function vertically by \(c\) scales its slope by \(c\).
  • Example: \(\frac{d}{dx}(5x^3) = 5 \cdot \frac{d}{dx}(x^3) = 5 \cdot (3x^2) = 15x^2\)

4. Sum/Difference Rule

  • The derivative of a sum (or difference) of functions is the sum (or difference) of their derivatives. $$ \frac{d}{dx}[f(x) \pm g(x)] = \frac{d}{dx}[f(x)] \pm \frac{d}{dx}[g(x)] = f'(x) \pm g'(x) $$
  • Intuition: Allows differentiating polynomials term by term.
  • Example: \(\frac{d}{dx}(x^2 + 5x - 3) = \frac{d}{dx}(x^2) + \frac{d}{dx}(5x) - \frac{d}{dx}(3) = 2x + 5 - 0 = 2x + 5\)

5. Product Rule

  • Used for finding the derivative of a product of two functions. $$ \frac{d}{dx}[f(x) g(x)] = f(x) g'(x) + g(x) f'(x) $$ "First times derivative of the second, plus second times derivative of the first."
  • Example: \(\frac{d}{dx}(x^2 \sin(x)) = x^2 (\cos(x)) + \sin(x) (2x)\)

6. Quotient Rule

  • Used for finding the derivative of a ratio (quotient) of two functions. $$ \frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{g(x) f'(x) - f(x) g'(x)}{[g(x)]^2} $$ "Low d-high minus high d-low, square the bottom and away we go!" (Mnemonic: d-high/d-low means derivative of numerator/denominator). Requires \(g(x) \neq 0\).
  • Example: \(\frac{d}{dx}\left(\frac{x}{x+1}\right) = \frac{(x+1)(1) - x(1)}{(x+1)^2} = \frac{1}{(x+1)^2}\)

7. Derivatives of Common Functions

  • Trigonometric:
    • \(\frac{d}{dx}(\sin x) = \cos x\)
    • \(\frac{d}{dx}(\cos x) = -\sin x\)
    • \(\frac{d}{dx}(\tan x) = \sec^2 x\)
  • Exponential & Logarithmic:
    • \(\frac{d}{dx}(e^x) = e^x\) (The exponential function \(e^x\) is its own derivative!)
    • \(\frac{d}{dx}(a^x) = a^x \ln(a)\) (where \(\ln\) is the natural logarithm)
    • \(\frac{d}{dx}(\ln x) = \frac{1}{x}\) (for \(x > 0\))
    • \(\frac{d}{dx}(\log_a x) = \frac{1}{x \ln a}\)

Connections to Other Topics

  • These rules are applied repeatedly when differentiating complex functions.
  • The Chain Rule is another crucial rule needed for composite functions (\(f(g(x))\)).
  • Form the basis for symbolic differentiation used in software packages (like SymPy, Mathematica).
  • Essential for finding partial derivatives in multivariable calculus (treat other variables as constants).

Summary

  • Differentiation rules provide shortcuts for finding derivatives without using the limit definition.
  • Key rules include: Constant, Power, Constant Multiple, Sum/Difference, Product, Quotient.
  • Knowing derivatives of common functions (trig, exponential, log) is necessary.
  • These rules streamline the process of finding rates of change and slopes.

Sources