The Quotient Rule
When functions are divided, we use the Quotient Rule:
\[\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}\]
A common mnemonic is: "Low d-High minus High d-Low, over the square of what's below."
Worked Examples
Example 1: Rational Function
Find the derivative of \(f(x) = \frac{x^2}{x+1}\).
- High: \(x^2\), d-High: \(2x\).
- Low: \(x+1\), d-Low: \(1\).
- Apply rule: \(\frac{(2x)(x+1) - (x^2)(1)}{(x+1)^2}\).
- Simplify: \(\frac{2x^2 + 2x - x^2}{(x+1)^2} = \frac{x^2 + 2x}{(x+1)^2}\).
- Result: \(f'(x) = \frac{x^2 + 2x}{(x+1)^2}\).
Example 2: Tangent via Quotient
Find the derivative of \(\tan x = \frac{\sin x}{\cos x}\).
- High: \(\sin x\), d-High: \(\cos x\).
- Low: \(\cos x\), d-Low: \(-\sin x\).
- Apply rule: \(\frac{(\cos x)(\cos x) - (\sin x)(-\sin x)}{\cos^2 x}\).
- Simplify: \(\frac{\cos^2 x + \sin^2 x}{\cos^2 x}\).
- Recall identity: \(\sin^2 x + \cos^2 x = 1\).
- Result is \(\frac{1}{\cos^2 x}\), which is \(\sec^2 x\).
- Result: \(\frac{d}{dx}(\tan x) = \sec^2 x\).
The Bridge to Quantum Mechanics
In Quantum Mechanics, we often deal with Normalization Factors. If a wavefunction \(\psi\) isn't normalized, we have to divide it by its total integral. When calculating how physical observables change, the Quotient Rule allows us to handle these ratios correctly, ensuring that probabilities always sum to 100%, even as the system evolves.