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 Δx (for "change in x) and Δy (for "change in y"). This makes the algebra appear more difficult, so here we use h for Δx 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.

Graph of tangent to curve

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)).

tangent delta

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.

slope of tangent explanation slope of tangent explanation

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 Δx):

`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 = 2x2+ 3x.

Example 2

a. Find `y’` from first principles if y = x2 + 4x.

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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