Derivatives from First Principles: The Limit Definition

What Is a Derivative?

The derivative of a function $f(x)$ at a point $x = a$ measures the instantaneous rate of change of $f$ at that point. Geometrically, it gives the slope of the tangent line to the curve at $(a, f(a))$ .

The core idea: start with the average rate of change over an interval, then shrink the interval to zero.

Learning Goals: By the end of this guide, you should be able to:

Explain the concept of a limit and how it relates to derivatives.

Differentiate simple functions from first principles.

Interpret the derivative as a rate of change and as a gradient.

Recognise when a function is not differentiable at a point.

The First Principles Definition

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

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

The expression $\frac{f(x+h) - f(x)}{h}$ is the difference quotient — the slope of the secant line through $(x, f(x))$ and $(x+h, f(x+h))$ .

As $h \to 0$ , the secant line pivots toward the tangent line, and the difference quotient approaches the derivative.

Worked Examples

Example 1: $f(x) = x^2$

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

So the gradient of $y = x^2$ at any point $x$ is $2x$ . At $x = 3$ , the tangent has slope $6$ .

Example 2: $f(x) = \frac{1}{x}$

f'(x) = \lim_{h \to 0} \frac{\frac{1}{x+h} - \frac{1}{x}}{h} = \lim_{h \to 0} \frac{x - (x+h)}{h \cdot x(x+h)} = \lim_{h \to 0} \frac{-1}{x(x+h)} = -\frac{1}{x^2}

Example 3: $f(x) = \sqrt{x}$

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

Multiply by the conjugate $\frac{\sqrt{x+h} + \sqrt{x}}{\sqrt{x+h} + \sqrt{x}}$ :

= \lim_{h \to 0} \frac{(x+h) - x}{h(\sqrt{x+h} + \sqrt{x})} = \lim_{h \to 0} \frac{1}{\sqrt{x+h} + \sqrt{x}} = \frac{1}{2\sqrt{x}}

From First Principles to Rules

The first principles method confirms the standard differentiation rules:

Function	Derivative	Obtained via first principles
$x^n$	$nx^{n-1}$	Binomial expansion
$\sin x$	$\cos x$	Using $\lim_{h\to 0}\frac{\sin h}{h} = 1$
$e^x$	$e^x$	Using the definition of $e$

Once proven, you can use these rules directly without re-deriving from limits each time.

When Does the Derivative Not Exist?

A function is not differentiable at a point where:

There is a sharp corner — the left and right limits of the gradient disagree (e.g., $|x|$ at $x = 0$ ).
There is a vertical tangent — the gradient approaches infinity (e.g., $\sqrt[3]{x}$ at $x = 0$ ).
There is a discontinuity — the function is not continuous at that point.

Common Mistakes

Forgetting to take the limit — Computing $\frac{f(x+h) - f(x)}{h}$ and stopping there gives the difference quotient, not the derivative.
Algebraic errors in expansion — Carefully expand $(x + h)^2$ , $(x + h)^3$ , etc. Missing a term is the most common source of error.
Dividing by zero — You must algebraically cancel the $h$ in the denominator before substituting $h = 0$ .

Exam Tips (A-Level / AP / IB)

First principles questions almost always involve $x^2$ , $x^3$ , $\frac{1}{x}$ or $\sqrt{x}$ . Master these four.
Show every algebraic step — marks are awarded for the process, not just the final answer.
If the question says "from first principles" or "using the definition", you must use the limit formula. Shortcut rules earn zero marks.

Frequently Asked Questions

Why can't I just use the power rule directly?

You can — once it's been proven! But understanding the limit definition is essential because it's the foundation of all differentiation rules. Exam boards specifically test this understanding.

What is the relationship between continuity and differentiability?

Differentiability implies continuity: if $f'(a)$ exists, then $f$ must be continuous at $a$ . But continuity does NOT imply differentiability — $|x|$ is continuous everywhere but not differentiable at $x = 0$ .

Applications of Differentiation — Once you can find the derivative, use it to solve optimisation and curve sketching problems.
Function Transformations — Understand how transforming $f(x)$ affects $f'(x)$ .
Sequences & Series — Limits are the shared foundation of both series convergence and differentiation.

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