How Do You Find the Derivative From First Principles Using Limits?

You find a derivative from first principles by forming the gradient of a secant line, simplifying its difference quotient, and then taking the limit as the horizontal change approaches zero.

This is IB Mathematics: Analysis and Approaches HL, sub-topic AHL 5.12: First principles, higher derivatives. The method below assumes a Paper 1-style analytic question.

The Limit Definition

For a function $f$, the derivative from first principles is

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

The quotient represents the gradient between the points ((x,f(x))) and $(x+h,f(x+h))$. As $h\to 0$, the second point approaches the first, so the secant gradient approaches the tangent gradient.

Here, $h$ is a non-zero horizontal increment before the limit is taken. The numerator is the corresponding change in $y$, so the quotient is an average rate of change. Algebraic simplification removes the factor causing the undefined expression at $h=0$; the limit then gives the instantaneous rate of change.

Worked Example

Find the derivative of $f(x)=x^2+3x$ from first principles.

First, substitute $x+h$ into the function:

$$f(x+h)=(x+h)^2+3(x+h).$$

Apply the definition:

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

Expand and simplify:

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

Now evaluate the limit:

$$f'(x)=2x+3.$$

As a quick check, at $x=2$, the derivative gives $f'(2)=7$. This matches the tangent gradient and confirms that the result depends on $x$, rather than being one fixed secant gradient.

The common misconception is to substitute $h=0$ immediately. This produces division by zero in the original quotient. You must first factor or simplify to cancel $h$, and only then evaluate the limit.

Exam Technique

For a command such as show or find from first principles, write the full limit definition, substitute correctly, simplify the numerator, cancel $h$, and evaluate the limit. Quoting the power rule alone will not answer the required method and will lose method marks.

On Paper 1, keep the algebra exact and show each stage clearly. Examiners look for the correct limiting expression and valid cancellation, not merely the final derivative.