\[f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\]

For \(y = f(x)\), the derivative can be written as:
• \(f'(x)\) (Lagrange notation)
• \(\frac{dy}{dx}\) (Leibniz notation)
• \(y'\) (prime notation)
• \(\frac{df}{dx}\)

\[\frac{d}{dx}(x^n) = nx^{n-1}\]

\[\frac{d}{dx}(ax^n) = anx^{n-1}\]

• Constant: \(\frac{d}{dx}(c) = 0\)
• Linear: \(\frac{d}{dx}(x) = 1\)
• Square: \(\frac{d}{dx}(x^2) = 2x\)
• Cubic: \(\frac{d}{dx}(x^3) = 3x^2\)

\[\frac{d}{dx}[cf(x)] = c \cdot f'(x)\]

\[\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)\]

\[\frac{d}{dx}[af(x) + bg(x)] = af'(x) + bg'(x)\]

\[\frac{d}{dx}(\sin x) = \cos x\]

\[\frac{d}{dx}(\cos x) = -\sin x\]

\[\frac{d}{dx}(\tan x) = \sec^2 x\]

\[\frac{d}{dx}(e^x) = e^x\]

\[\frac{d}{dx}(\ln x) = \frac{1}{x}\]

\[\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}\]

\[\frac{d}{dx}[f(g(x))] = f'(g(x)) \times g'(x)\]

• Function inside another function (composition)
• Examples: \((3x+2)^5\), \(\sin(2x)\), \(e^{x^2}\), \(\ln(x^2+1)\)

\[\frac{d}{dx}[u \cdot v] = u \frac{dv}{dx} + v \frac{du}{dx}\]

\[\frac{d}{dx}[f(x) \cdot g(x)] = f(x) \cdot g'(x) + g(x) \cdot f'(x)\]

• Multiplication of two functions
• Examples: \(x^2 \sin x\), \(e^x \cos x\), \(x \ln x\)

\[\frac{d}{dx}\left[\frac{u}{v}\right] = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}\]

"Low dee-high minus high dee-low, square the bottom and away we go!"

• Division of two functions
• Examples: \(\frac{x^2}{x+1}\), \(\frac{\sin x}{x}\), \(\frac{e^x}{x^2}\)

\[f''(x) = \frac{d^2y}{dx^2} = \frac{d}{dx}\left[\frac{dy}{dx}\right]\]

• Measures concavity of the graph
• Related to acceleration if y represents position
• Used to determine nature of stationary points

\[m_{\text{tangent}} = f'(a)\]

\[y - f(a) = f'(a)(x - a)\]

\[m_{\text{normal}} = -\frac{1}{f'(a)}\]

\[y - f(a) = -\frac{1}{f'(a)}(x - a)\]

\[f'(x) = 0\]

At stationary point \(x = a\) where \(f'(a) = 0\):
• If \(f''(a) > 0\): Local minimum
• If \(f''(a) < 0\): Local maximum
• If \(f''(a) = 0\): Test inconclusive (use first derivative test)

Check sign of \(f'(x)\) before and after \(x = a\):
• \(+ \to -\): Local maximum
• \(- \to +\): Local minimum
• \(+ \to +\) or \(- \to -\): Point of inflection

1. Identify the quantity to maximize/minimize
2. Express it as a function of one variable
3. Find \(f'(x) = 0\) to locate stationary points
4. Use second derivative test or boundary analysis
5. Verify it's a maximum/minimum (not just stationary point)

• Maximizing area or volume
• Minimizing cost, distance, or time
• Finding optimal dimensions

When two or more variables are related and change with time, their rates of change are also related

1. Write equation relating the variables
2. Differentiate both sides with respect to time
3. Substitute known values and solve for unknown rate

\[\frac{dx}{dt}, \frac{dy}{dt}\]