Derivative from First Principles

The derivative from first principles is the limit of secant slopes as the second point approaches the first.

A derivative is the limiting slope of nearby secant lines. A smooth curve has a highlighted point, a tangent line there, and a nearby secant line. A small delta-x bracket shows how the secant approaches the tangent as the second point moves closer.

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

\frac{f(a+h)-f(a)}{h},\qquad h\ne0.

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

f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h},

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.

Worked example: derive the power pattern

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

Evaluating the derivative

At $x=3$,

$$ f'(3)=2(3)=6. $$

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

$$ y-9=6(x-3). $$

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

Another algebraic example

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

\frac{f(x+h)-f(x)}h =\frac{3(x+h)+5-(3x+5)}h =\frac{3h}h=3.

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

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.