Differential Calculus — Notes

DIFFERENTIAL CALCULUS

Definitions, formulas, and worked examples

1. The Gradient Function

Consider a curve whose equation is given by \(y = f(x)\).

The gradient of the secant \(AB\) is:

\( \dfrac{f(x + \Delta x) - f(x)}{\Delta x} \)

Let \(\Delta x = h\) where \(h \to 0\). As \(B\) moves closer to \(A\), the slope of the secant approaches the slope of the tangent.

The limiting value is the gradient function of \(y = f(x)\):

\( \displaystyle \lim_{\Delta x \to 0} \dfrac{f(x + \Delta x) - f(x)}{\Delta x} \)

2. The Derivative of a Function

For the curve \(y = f(x)\), the expression

\( \dfrac{\Delta y}{\Delta x} = \dfrac{f(x + \Delta x) - f(x)}{\Delta x} \)

is the gradient of the secant \(AB\). As \(B\) approaches \(A\), we define

\( \displaystyle m_t = \lim_{\Delta x \to 0} \dfrac{f(x + \Delta x) - f(x)}{\Delta x} = \dfrac{dy}{dx} \)

Thus, \(\dfrac{dy}{dx}\) is the derivative of \(y = f(x)\).

3. Meaning of the Derivative

A small increment in \(x\) (\(\Delta x\)) causes a small increment in \(y\) (\(\Delta y\)).

\( \Delta y = f(x + \Delta x) - f(x) \)

The ratio \(\dfrac{\Delta y}{\Delta x}\) is the slope of the secant. As \(\Delta x \to 0\):

\( \displaystyle \dfrac{dy}{dx} = \lim_{\Delta x \to 0} \dfrac{\Delta y}{\Delta x} \)

4. Interpretation and Notation

\(\dfrac{dy}{dx}\) is read “dee y by dee x” and represents the limiting slope of the tangent. The process is called differentiation.

\(f'(x)\) — “f-prime of x”
\(\dfrac{df}{dx}\) — “dee f by dee x”
\(\dfrac{d}{dx}[f(x)]\) — operator form

If \(y = f(x)\), then \(\dfrac{dy}{dx} = f'(x)\).

It represents the gradient function or rate of change of \(y\) with respect to \(x\).

5. Differentiation from First Principles

The derivative from first principles is defined as:

\( \displaystyle f'(x) = \lim_{h \to 0} \dfrac{f(x + h) - f(x)}{h} \)

Example 1

Find the derivative of \(y = x^2\) from first principles.

\( \begin{aligned} f(x) &= x^2,\\ f(x+h) &= (x+h)^2 = x^2 + 2xh + h^2,\\ f(x+h) - f(x) &= 2xh + h^2,\\ \dfrac{f(x+h) - f(x)}{h} &= 2x + h.\\ \end{aligned} \)
As \(h \to 0\): \( f'(x) = 2x. \)

Example 2

Find the derivative of \(y = x^3\) from first principles.

\( \begin{aligned} f(x) &= x^3,\\ f(x+h) &= (x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3,\\ f(x+h) - f(x) &= 3x^2h + 3xh^2 + h^3,\\ \dfrac{f(x+h) - f(x)}{h} &= 3x^2 + 3xh + h^2.\\ \end{aligned} \)
As \(h \to 0\): \( f'(x) = 3x^2. \)

Summary

The derivative represents the slope or rate of change at a point.
From first principles:
\( \displaystyle f'(x) = \lim_{h \to 0} \dfrac{f(x + h) - f(x)}{h} \)
If \(y = x^n\), then \(\dfrac{dy}{dx} = nx^{n-1}\).
Differentiation gives the gradient of a function at any point.

CALCULUS