Differentiation

Differential calculus concerns the rate of change of a function with respect to the change in the variable on which it depends.

Speed is the distance travelled by an object in a unit period of time. Instantaneous speed is the speed at a particular moment. Average speed is total distance divided by total time.

Velocity is speed in a particular direction. Acceleration measures how velocity changes with time.

The derivative of a function at a particular value measures the rate at which the function is changing at that value.

The derivative at a point is also the gradient of the tangent to the graph at that point.

Finding Gradients of Curves

The gradient of a straight line is found by dividing the change in the y-axis by the change in the x-axis.

But what happens if we need the gradient of a curve?

The points \(A(2,4)\) and \(B(3,9)\) lie on the curve \(y = x^2\).

The average gradient from A to B is the gradient of the chord AB.

\[ m_{AB} = \frac{y_{2}-y_{1}}{x_{2}-x_{1}} \] \[ m_{AB} = \frac{9-4}{3-2} \] \[ m_{AB}=5 \]

\[ \Delta y = \text{ change in } y \] \[ \Delta x = \text{ change in } x \] \[ m_{AB} = \frac{y_{2}-y_{1}}{x_{2}-x_{1}} = \frac{\Delta y}{\Delta x} \]

As B moves towards A, the gradient changes.

\[ m_{AB} = \frac{y_{2}-y_{1}}{x_{2}-x_{1}} \] \[ m_{AB} = \frac{6.25 - 4}{2.5 - 2} \] \[ m_{AB} = \frac{2.25}{0.5} \] \[ m_{AB} = 4.5 \qquad \frac{\Delta y}{\Delta x} = 4.5 \]

\[ m_{AB} = \frac{y_{2}-y_{1}}{x_{2}-x_{1}} \] \[ m_{AB} = \frac{4.41 - 4}{2.1 - 2} \] \[ m_{AB} = \frac{0.41}{0.1} \] \[ m_{AB} = 4.1 \qquad \frac{\Delta y}{\Delta x} = 4.1 \]

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

The closer B is to A, the nearer the gradient reaches a limit.

Graph of x^2 with tangent — \(f(x) = x^2\)

This limit is the gradient of the curve and occurs at the tangent to the curve at the point A..

When \(A = (2,4)\), the gradient is 4.

Other points have different gradients.

Is there an easier way to find the gradient at any point?

First Principles

Let \(A(x, f(x))\) and \(B(x+h, f(x+h))\) be points on the graph \(y=f(x)\).

\[ m_{AB} = \frac{y_{2}-y_{1}}{x_{2}-x_{1}} \] \[ m_{AB} = \frac{f(x+h)-f(x)}{(x+h)-x} \] \[ m_{AB} = \frac{f(x+h)-f(x)}{h} \]

\[ m_{AB} = \frac{y_{2}-y_{1}}{x_{2}-x_{1}} \] \[ m_{AB} = \frac{\Delta y}{\Delta x} = \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x} \] \[ m_{AB} = \frac{dy}{dx} \quad\text{(Leibniz notation)} \]

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

\[ \text{The gradient is derived by reducing the difference between } x \text{ and } h. \] \[ \text{As } h \text{ tends to zero, the gradient tends to a limit.} \] \[ \text{This limit is the gradient of the tangent to the curve at point A.} \] \[ f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h} \] \[ f'(x) \text{ is called the derivative or derived function of } f(x). \] \[ \text{It is the rate of change of the function and the gradient of the tangent to its graph.} \]

\[ \text{Leibniz notation can also be used.} \] \[ \text{Instead of writing } f'(x),\ \text{write } \frac{dy}{dx}. \] \[ \frac{dy}{dx} = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h} \]

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

Deriving \(f'(x)\) from \(f(x)\)

Using the definition:

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

Example

If \(f(x)=x^2\), find \(f'(x)\).

\[ \begin{aligned} f'(x) &= \lim_{h\to 0} \frac{(x+h)^2 - x^2}{h} \\ &= \lim_{h\to 0} \frac{x^2 + 2xh + h^2 - x^2}{h} \\ &= \lim_{h\to 0} \frac{2xh + h^2}{h} \\ &= \lim_{h\to 0} (2x + h) \\ &= 2x \end{aligned} \]

\[ \begin{aligned} f'(x) &= \lim_{h\to 0} \frac{(x+h)^2 - x^2}{h} \\ &= \lim_{h\to 0} \frac{x^2 + 2xh + h^2 - x^2}{h} \\ &= \lim_{h\to 0} (2x + h) \\ &= 2x \end{aligned} \]

The derived function is \(f'(x)=2x\). At \(x=3\), the derivative is 6.

This means that the rate of change of the function is 2x, and that the gradient of the tangent to the graph of the curve y = x² at any point is found by doubling the x co-ordinate.

The gradient of the tangent to the graph at the point (3, 9) is 6.
The gradient of the tangent to the graph at the point (4, 16) is 8.

The equation of the tangent can be found by using y - b = m(x-a)

In General

If \(f(x)=x^n\), then \(f'(x)=nx^{n-1}\) (for rational \(n\)).

Example

If \(f(x)=x^3\), find \(f'(x)\).

\[ f'(x) = 3x^{2} \]

More examples

Books

Printed resources available at Amazon

Differentiation by First Principles

Differentiation by First Principles (Calculus Revision)

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These notes are designed as a revision aid for differentiation by first principles, covering core rules and trigonometric derivatives derived from limits.

Derivative of \(f(x) = g(x) + h(x)\)
Derivative of \(f(x) = ax^n\)
Derivative of \(f(x) = (x + a)^n\)
Derivative of \(f(x) = (ax + b)^n\)
The Chain Rule
Finding trigonometric derivatives by first principles
Derivative of \(f(x) = \sin x\)
Derivative of \(f(x) = \sin(ax)\)
Derivative of \(f(x) = \cos x\)
Derivative of \(f(x) = \tan x\)
Product Rule
Quotient Rule

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