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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".

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 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:

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, can be written:

So also, the slope PQ will be given by:

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 will approach the required slope.

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

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):

You will also come across the following for delta method:

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):

or f ’(x) or y’.

Example 1:

Find from first principles if y = 2x²+ 3x.

Solution:

f(x) = 2x²+ 3x so

We now need to find:

We have found an expression that can give us the slope of the tangent anywhere on the curve.

If x = -2, the slope is 4(-2) + 3 = -5 (red, in the graph below)

If x = 1, the slope is 4(1) + 3 = 7 (green)

If x = 4, the slope is 4(4) + 3 = 19 (black)

We can see that our answers are correct when we graph the curve (which is a parabola) and observe the slopes of the tangents.

This is what makes calculus so powerful. We can find the slope anywhere on the curve (i.e. the rate of change of the function anywhere).

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.

Solution:

a. Note: y' means "the first derivative". This can also be written dy/dx.

Now f(x) = x² + 4x

So

b. When x = 1, m = 2(1) + 4 = 6

When x = -6, m = 2(-6) + 4 = -8

c. Sketch:

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LiveMath is not really designed to do this type of operation. But here it is, anyway. It only does the first part of the problem that we just did.

LIVEMath

 

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Chapter Contents

• Differentiation
• 1. Limits and Differentiation
• 2. The Slope of a Tangent to a Curve (Numerical)
• 3. The Derivative from First Principles
• 4. Derivative as an Instantaneous Rate of Change
• 5. Derivatives of Polynomials
• 6. Derivatives of Products and Quotients
• 7. Differentiating Powers of a Function
• Differentiation Java applet
• Derivative of a Cubic - Exploration using GeoGebra
• 8. Differentiation of Implicit Functions
• 9. Higher Derivatives
• 10. Partial Derivatives

• Comments, Questions?

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