7.1 Rules of Differentiation

Differentiation rules are the building blocks of calculus, allowing us to find rates of change for various functions. These rules, including the power rule, product rule, quotient rule, and chain rule, give us tools to tackle complex mathematical problems.

By combining these rules, we can differentiate even the most intricate functions. Understanding how to apply these rules in different scenarios is crucial for solving real-world problems involving rates of change, optimization, and more.

Differentiation Rules

Power rule for polynomial differentiation

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• Power rule: $\frac{d}{dx}x^n = nx^{n-1}$ multiplies coefficient by exponent and subtracts 1 from exponent ($x^3 \rightarrow 3x^2$)
• Constant rule: $\frac{d}{dx}c = 0$ states derivative of constant is always 0 ($5 \rightarrow 0$)
• Linearity of differentiation: $\frac{d}{dx}(af(x) + bg(x)) = a\frac{d}{dx}f(x) + b\frac{d}{dx}g(x)$ differentiates each term separately, multiplies each derivative by its coefficient, and adds results ($3x^2 + 2x \rightarrow 6x + 2$)

Product rule for function derivatives

• Product rule: $\frac{d}{dx}(f(x)g(x)) = f(x)\frac{d}{dx}g(x) + g(x)\frac{d}{dx}f(x)$ finds derivative by:
  1. Multiplying first function by derivative of second function
  2. Multiplying second function by derivative of first function
  3. Adding the two resulting terms together
• Example: $\frac{d}{dx}(x^2 \cdot \sin x) = x^2 \cdot \cos x + \sin x \cdot 2x$

Quotient rule for function ratios

• Quotient rule: $\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{g(x)\frac{d}{dx}f(x) - f(x)\frac{d}{dx}g(x)}{[g(x)]^2}$ finds derivative by:
  1. Multiplying denominator function by derivative of numerator function
  2. Multiplying numerator function by derivative of denominator function
  3. Subtracting second term from first term
  4. Dividing result by square of denominator function
• Example: $\frac{d}{dx}\left(\frac{x^2}{x+1}\right) = \frac{(x+1)(2x) - x^2(1)}{(x+1)^2}$

Chain rule for composite functions

• Chain rule: $\frac{d}{dx}f(g(x)) = f'(g(x))g'(x)$ finds derivative by:
  1. Finding derivative of outer function, $f$, with respect to its input
  2. Finding derivative of inner function, $g$, with respect to $x$
  3. Multiplying the two derivatives together
• Generalized chain rule: $\frac{d}{dx}f(g(h(x))) = f'(g(h(x)))g'(h(x))h'(x)$ applies chain rule successively for each nested function and multiplies derivatives of all functions in composition
• Example: $\frac{d}{dx}\sin(x^2) = \cos(x^2) \cdot 2x$

Application of Differentiation Rules

Combine differentiation rules to find derivatives of complex functions

• Identify structure of complex function as sum, product, quotient, or composition of functions
• Apply appropriate differentiation rules based on function's structure:
  • Linearity of differentiation for sums of functions
  • Product rule for products of functions
  • Quotient rule for ratios of functions
  • Chain rule for composite functions
• Simplify resulting derivative expression if necessary
• Example: $\frac{d}{dx}\left(\frac{x^3}{\sqrt{x+1}}\right) = \frac{(x+1)^{1/2}(3x^2) - x^3\left(\frac{1}{2}(x+1)^{-1/2}\right)}{(x+1)}$