AcademyDifferentiation
Academy
Derivatives
Level 1 - Math I (Physics) topic page in Differentiation.
Derivatives
The derivative of a function \(f(x)\) measures the rate of change of \(f\) with respect to \(x\).
Definition
\[f'(x) = \frac{df}{dx} = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\]
Notation
There are several common notations for derivatives:
- Leibniz notation: \(\frac{dy}{dx}\)
- Lagrange notation: \(f'(x)\)
- Euler notation: \(Df\)
Basic Rules
Constant Multiple
\[\frac{d}{dx}[cf(x)] = cf'(x)\]
Sum Rule
\[\frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)\]
Product Rule
\[\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)\]
Quotient Rule
\[\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}\]
The Power Rule
Power Rule
\[\frac{d}{dx}[x^n] = nx^{n-1}\]
This works for any real exponent \(n\).