Calculus: Derivatives of Functions

Derivative of the Reciprocal of a Function

• The derivative of the reciprocal of a function $f(x)$ is given by:
  • $\frac{d}{dx} \left(\frac{1}{f(x)}\right) = -\frac{f'(x)}{[f(x)]^2}$
• Derivation involves calculating the limit of the incremental ratio:
  • $\lim_{h \to 0}\frac{\frac{1}{f(x+h)} - \frac{1}{f(x)}}{h}$
• Simplifying by common denominators yields:
  • $\lim_{h \to 0} \frac{f(x) - f(x+h)}{h} * \frac{1}{f(x+h)f(x)}$
• This limit splits into two parts:
  • $\lim_{h \to 0} \frac{f(x) - f(x+h)}{h}$ with a limit of $-f'(x)$ as $h$ approaches $0$
  • $\lim_{h \to 0} \frac{1}{f(x+h)f(x)}$ with a limit of $\frac{1}{[f(x)]^2}$ as $h$ approaches $0$
• Multiplying the two limits produces the original formula.

Derivative of the Quotient of Two Functions

• The derivative of the quotient of two functions $p(x)$ and $q(x)$ is given by:
  • $\frac{d}{dx} \left(\frac{p(x)}{q(x)}\right)= \frac{p'(x)q(x) - p(x)q'(x)}{[q(x)]^2}$
• This formula derives from viewing the quotient as a product:
  • $\frac{p(x)}{q(x)} = p(x) * \frac{1}{q(x)}$
• Using the product rule and the reciprocal rule, the derivative of the quotient is obtained.

Examples

• Reciprocal Examples:
  • $\frac{d}{dx} \left(\frac{1}{\cos(x)}\right) = \frac{\sin(x)}{\cos^2(x)}$
  • $\frac{d}{dx} \left(\frac{1}{\ln(x)}\right) = -\frac{1}{x\ln^2(x)}$
• Quotient Examples:
  • $\frac{d}{dx} \left(\frac{\sin(x)}{\cos(x)}\right) = \frac{\cos^2(x) + \sin^2(x)}{\cos^2(x)} = \frac{1}{\cos^2(x)}$
  • $\frac{d}{dx} \left(\frac{e^x}{x}\right) = \frac{xe^x - e^x}{x^2}$