# 3. The Derivative from First Principles

In this section, we will differentiate a function from "first principles". This means we will start from scratch and use algebra to find a general expression for the slope of a curve, at any value *x*.

First
principles is also known as "delta
method", since many texts use Δ*x* (for "change in *x*) and Δ*y* (for "change in *y*"). This makes the algebra appear more difficult, so here we
use *h* for Δ*x* instead. We still call it "delta
method".

### NOTE

If you want to see how to find slopes (gradients) of tangents directly using derivatives, rather than from first principles, go to Tangents and Normals in the Applications of Differentiation chapter.

We wish to find an **algebraic method** to find the slope
of *y* = *f*(*x*) at *P*, to save
doing the numerical substitutions that we saw in the last section (Slope of a Tangent to a Curve - Numerical Approach).

We can approximate this value by taking a point somewhere near to
*P*(*x*, *f*(*x*)), say *Q*(*x* +
*h*, *f*(*x* + *h*)).

The value `g/h` is an approximation to the slope of the tangent which we require.

We can also write this slope as `("change in"\ y) /("change in"\ x)` or:

`m=(Deltay)/(Deltax`

If we move *Q* closer and closer to *P*, the line *PQ* will get closer and closer to the tangent at *P* and so the slope of *PQ* gets closer to the slope that we want.

If we let *Q* go all the way to touch *P* (i.e. `h = 0`), then we would have the **exact** slope of the tangent.

Now, `g/h` can be written:

`g/h=(f(x+h)-f(x))/h`

So also, the slope *PQ* will be given by:

`m=(y_2-y_1)/(x_2-x_1)=(Deltay)/(Deltax)` `=(f(x+h)-f(x))/h`

But we require the slope **at** *P*, so we let `h → 0` (that is let *h* approach `0`), then in effect, *Q* will
approach *P* and `g/h` will approach the
required slope.

## The Slope of a Curve as a Derivative

Putting this together, we can write the slope of the tangent at *P* as:

`dy/dx=lim_(h->0)(f(x+h)-f(x))/h`

This is called **differentiation from first principles,** (or
the **delta method**). It gives the instantaneous rate of change
of *y *with respect to
*x.*

This is equivalent to the following (where before we were using *h* for Δ*x*):

`dy/dx=lim_(Deltax->0)(Deltay)/(Deltax`

You will also come across the following way of writing the Delta Method:

`dy/dx=lim_(Deltax->0)(f(x+Deltax)-f(x))/(Deltax`

## Notation for the Derivative

IMPORTANT: The **derivative** (also called **differentiation**) can be written in several ways. This can cause some confusion when we first learn about differentiation.

The following are equivalent ways of writing the first derivative of `y = f(x)`:

`dy/dx` or `f’(x)` or `y’`.

### Example 1

Find `dy/dx` from
first principles if *y *=
2*x*^{2}+
3*x*.

### Example 2

a. Find `y’` from first
principles if *y *=* x*^{2 }+
4*x*.

b. Find the slope of the tangent where *x*
= 1 and also where *x *= −6.

c. Sketch the curve and both tangents.

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