Differentiation from First Principles

The Limit Definition of the Derivative

The derivative of a function $f(x)$ is defined as:

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

This expression is called the difference quotient. The numerator $f(x + h) - f(x)$ represents the change in the function's value when $x$ increases by a small amount $h$. Dividing by $h$ gives the average rate of change over the interval $[x, x + h]$. Taking the limit as $h \to 0$ transforms this average rate of change into the instantaneous rate of change — the derivative.

An equivalent notation uses $\delta x$ (a small increment in $x$):

$\frac{dy}{dx} = \lim_{\delta x \to 0} \frac{f(x + \delta x) - f(x)}{\delta x}$

Both notations are used across exam boards, so you should be comfortable with either.

Geometric Interpretation

Consider the curve $y = f(x)$. Take two points on the curve: $P = (x, f(x))$ and $Q = (x + h, f(x + h))$. The line through $P$ and $Q$ is a secant line, and its gradient is:

$m_{PQ} = \frac{f(x + h) - f(x)}{h}$

As $h \to 0$, the point $Q$ slides along the curve towards $P$, and the secant line rotates towards the tangent line at $P$. The gradient of the tangent is therefore the limit of the secant gradient — which is precisely the derivative $f'(x)$.

This geometric picture is essential for understanding what differentiation actually means. The derivative at a point is the gradient of the tangent to the curve at that point.

Deriving the Power Rule from First Principles

The power rule states that if $f(x) = x^n$, then $f'(x) = nx^{n-1}$. We can prove this for positive integer values of $n$ using first principles.

Proof for $f(x) = x^2$:

$f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - x^2}{h}$

Expanding $(x + h)^2 = x^2 + 2xh + h^2$:

$f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - x^2}{h} = \lim_{h \to 0} \frac{2xh + h^2}{h}$

Factorising $h$ from the numerator:

$f'(x) = \lim_{h \to 0} (2x + h) = 2x$

This confirms the power rule: if $f(x) = x^2$, then $f'(x) = 2x$.

Proof for $f(x) = x^3$:

$f'(x) = \lim_{h \to 0} \frac{(x + h)^3 - x^3}{h}$

Using the binomial expansion $(x + h)^3 = x^3 + 3x^2h + 3xh^2 + h^3$:

$f'(x) = \lim_{h \to 0} \frac{3x^2h + 3xh^2 + h^3}{h} = \lim_{h \to 0} (3x^2 + 3xh + h^2) = 3x^2$

Again, this matches $nx^{n-1}$ with $n = 3$.

Differentiating Linear and Constant Functions

For $f(x) = mx + c$ (a linear function):

$f'(x) = \lim_{h \to 0} \frac{m(x + h) + c - (mx + c)}{h} = \lim_{h \to 0} \frac{mh}{h} = m$

The derivative of a linear function is its gradient — exactly as expected.

For $f(x) = c$ (a constant function):

$f'(x) = \lim_{h \to 0} \frac{c - c}{h} = \lim_{h \to 0} 0 = 0$

Constant functions have zero gradient everywhere, so their derivative is zero.

First Principles with Coefficient and Sum Functions

For $f(x) = ax^n$ where $a$ is a constant, you can show from first principles that $f'(x) = anx^{n-1}$. The constant $a$ factors out of the limit.

For a sum $f(x) = g(x) + k(x)$, the limit of a sum equals the sum of the limits, so:

$f'(x) = g'(x) + k'(x)$

This is the foundation for differentiating polynomials term by term.

Tip 1: Follow a Systematic Process

Always write out the full definition first: $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$. Then substitute, expand, simplify, cancel $h$, and finally take the limit. Skipping steps is where most errors occur.

Tip 2: Expand Brackets Carefully

The most common source of errors is incorrect expansion of $(x + h)^n$. For $n = 2$, remember $(x+h)^2 = x^2 + 2xh + h^2$. For $n = 3$, use Pascal's triangle or the binomial theorem. Always double-check your expansion before proceeding.

Tip 3: Factor Out $h$ Before Taking the Limit

After simplifying the numerator, every term should contain $h$ as a factor (if it doesn't, you've made an error — the $f(x)$ terms should cancel). Factor out $h$, cancel it with the $h$ in the denominator, and only then let $h \to 0$.

Tip 4: Show Your Limit Notation

Exam boards award marks for correct use of the $\lim_{h \to 0}$ notation. Keep writing it at each stage until you actually evaluate the limit. Dropping it early can cost you method marks.

Tip 5: Verify Your Answer

After finding the derivative from first principles, check it against the power rule. If you differentiated $f(x) = 3x^2$ and got $f'(x) = 6x$, that matches $2 \times 3x^{2-1} = 6x$. This quick check can catch errors.