Derivative from First Principles

The derivative is not just a rule sheet—it begins as the limiting slope of shrinking secant intervals.

From average to instantaneous change

Between $x=a$ and $x=a+h$, average rate of change is

As $h$ approaches $0$, the second point approaches the first. If the secant slopes approach a finite common value, that value is the instantaneous rate and tangent slope.

Definition at a point

The derivative of $f$ at $x=a$ is

provided the limit exists. The fraction is called the difference quotient.

We never substitute $h=0$ into the unsimplified quotient because that would divide by zero. We simplify for nonzero $h$, then take the limit.

Derivative as a function

Replacing $a$ with a variable $x$ gives

This produces a new function whose output is the original function's instantaneous rate at each differentiable input.

Worked example: derive the power pattern

Thus the tangent slope to $y=x^2$ at input $x$ is $2x$.

Evaluating the derivative

At $x=3$,

The point on the curve is $(3,9)$ and the tangent equation is

The derivative value and the function value describe different features: slope versus height.

Another algebraic example

For $f(x)=3x+5$,

Therefore $f'(x)=3$, matching the constant slope of the line.

When the derivative does not exist

A derivative can fail at a corner, cusp, vertical tangent, discontinuity, or point with different one-sided slope limits. For $f(x)=|x|$ at $0$, left secant slopes approach $-1$ while right secant slopes approach $1$.

Continuity is necessary for differentiability, though continuity alone is not sufficient.

Alternative limit form

An equivalent definition is

Here $(x,f(x))$ approaches $(a,f(a))$. Both forms describe the same secant-to-tangent process.

Units and meaning

If $f$ measures metres and $x$ measures seconds, $f'(a)$ has units metres per second. If cost depends on production quantity, the derivative has dollars per additional item near that production level.

Always interpret the sign, units, and input location.

Why first principles still matters

Differentiation rules make routine work faster, but first principles explains why a derivative is a rate and slope. It also provides a way to test differentiability and derive rules rather than treating them as magic.

Common mistakes

Using $f(x)+h$ instead of $f(x+h)$. Replace every $x$ in the formula with $x+h$.

Substituting $h=0$ before simplifying. The original quotient is undefined there.

Expanding $(x+h)^2$ incorrectly. Include the middle term $2xh$.

Confusing $f'(a)$ with $f(a)$. One is a rate; the other is an output.

Assuming every continuous point is differentiable. Corners can be continuous but not differentiable.

Quick self-check

• Is the difference quotient written exactly?
• Did I compute $f(x+h)$ with parentheses?
• Can a factor of $h$ be exposed and cancelled for $h\ne0$?
• Did I take the limit only after simplification?
• Does the derivative's sign, value, and unit fit the graph or context?
• Do left and right slope limits agree?

Related topics

• Limits
• Instantaneous Rate of Change
• Basic Differentiation Rules
• Tangent Lines