Introduction & First Principles

Video lesson · from 0:261 worked example

One lesson video covers all of Differentiation, so it opens at 0:26 for this topic — not from the beginning.

Theory

Calculus provides a tool for solving problems involving motion, such as orbits of planets or paths of rockets. It can be used to study any situation where a rate of increase or decrease is involved.

The instantaneous speed at a certain moment in time is known as the rate of change. This is the same as finding the gradient of the tangent to a curve at a certain point.

Differentiation from First Principles

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

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

$f'(x)$ is the derived function or derivative of $f(x)$ .

⚠️ Common Examiner Traps

Keep the limit notation until the end: every line needs $\lim_{h \to 0}$ in front of it until you actually take the limit. Dropping it early is a communication error even if the algebra is right.
Expand $f(x+h)$ carefully: $(x+h)^2 = x^2 + 2xh + h^2$ , not $x^2 + h^2$ . This single expansion is where most first-principles marks are lost.
Cancel the $h$ , do not set it to zero: divide the numerator and denominator by $h$ first. Substituting $h = 0$ before cancelling gives $\tfrac{0}{0}$ .
Answer the question asked: "differentiate from first principles" means marks are for the process. Quoting the power rule scores nothing.

Worked examples

Example 1

Prove that if $f(x) = x^2$ then $f'(x) = 2x$ using differentiation from first principles.

\begin{aligned} f'(x) &= \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} \\ &= \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 \text{ as required.} \end{aligned}