2.1 Limit and Its Relation to Differentiation
Differentiation is the mathematical process of finding the rate of change of a function with respect to a variable.
Concept of Limit
The limit of a function $f(x)$ as $x$ approaches $a$ is written as $\lim_{x \to a} f(x) = L$.
First Derivative from First Principles
The gradient of a curve $y = f(x)$ at a point is obtained by finding the limit of the secant gradient as $\delta x \to 0$:
$$\frac{dy}{dx} = \lim_{\delta x \to 0} \frac{\delta y}{\delta x} = \lim_{\delta x \to 0} \frac{f(x + \delta x) - f(x)}{\delta x}$$
2.2 The First Derivative
Basic Differentiation Rules
For algebraic functions where $a$ and $n$ are real numbers:
- Constant Rule: $\frac{d}{dx}(a) = 0$
- Power Rule: $\frac{d}{dx}(ax^n) = anx^{n-1}$
- Sum/Difference Rule: $\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$
Advanced Differentiation Rules
- Composite Functions (Chain Rule): If $y = f(u)$ and $u = g(x)$, then:
$$\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$$
For $y = [u(x)]^n$:
$$\frac{dy}{dx} = n[u(x)]^{n-1} \cdot u'(x)$$
- Product Rule: If $y = u \cdot v$, then:
$$\frac{dy}{dx} = u\frac{dv}{dx} + v\frac{du}{dx}$$
- Quotient Rule: If $y = \frac{u}{v}$, then:
$$\frac{dy}{dx} = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$$
2.3 The Second Derivative
The second derivative is obtained by differentiating the first derivative $\frac{dy}{dx}$ with respect to $x$:
$$\text{Second Derivative} = \frac{d}{dx}\left(\frac{dy}{dx}\right) = \frac{d^2y}{dx^2} = f''(x)$$
2.4 Application of Differentiation
Tangent and Normal to a Curve
- Gradient of Tangent ($m_t$): $m_t = \left.\frac{dy}{dx}\right|_{x = x_1}$
- Equation of Tangent: $y - y_1 = m_t(x - x_1)$
- Gradient of Normal ($m_n$): Since tangent and normal are perpendicular, $m_t \cdot m_n = -1 \implies m_n = -\frac{1}{m_t}$
- Equation of Normal: $y - y_1 = m_n(x - x_1)$
Turning Points and Stationary Points
At a stationary point, $\frac{dy}{dx} = 0$. Nature of stationary points can be determined using the second derivative test:
- Maximum Point: $\frac{dy}{dx} = 0$ and $\frac{d^2y}{dx^2} < 0$
- Minimum Point: $\frac{dy}{dx} = 0$ and $\frac{d^2y}{dx^2} > 0$
- Point of Inflexion: $\frac{d^2y}{dx^2} = 0$ (requires sign test of $\frac{dy}{dx}$ around the point)
Rates of Change and Small Changes / Approximations
- Connected Rates of Change: Using Chain Rule, $\frac{dy}{dt} = \frac{dy}{dx} \times \frac{dx}{dt}$
- Small Changes and Approximations: $\delta y \approx \frac{dy}{dx} \cdot \delta x$ and new value $y + \delta y \approx y + \frac{dy}{dx} \cdot \delta x$
- Percentage Change: Percentage change in $x = \frac{\delta x}{x} \times 100\%$