4 Basic differentiation

Differentiation is all about calculating the slope or gradient of a curve y(x), at a given point, x.


The gradient is the

Think of road signs: a 1-in-10 hill means you travel 1 metre upwards for every 10 metres you travel along.

Notation: We use the symbol , Δ delta, to mean a (large) change in the value of a variable. If, say, x changes from a value of x1, to a new value, x2, then


Δx = x2 - x1

So the gradient of a curve y(x) can be written as:

gradient =

Linear Equations


For a straight line graph of equation y(x) = mx + c, the gradient is given simply by the value of m.

Examples

Measuring gradients

If we don't know the equation of the straight line, we can work out the gradient by tabulating the values of y vs. x and plotting the graph.

Example Values of y and x are given below, what is the gradient?
x-3-2 -101 23
y-11-8 -5-21 47

Graphical Method

(i) Plot graph



(ii) Choose any 2 points along the line (x1,y1) and (x2,y2)

(iii) Draw the triangles (as in the diagram), or just calculate x and y.

(iv) Calculate gradient from: gradient =

Numerical Method: choose the points we have values for, say, (-2, -8) and (1,1). We now have:

gradient = = = 3

Since the intercept is at y = -2, we know that the eqn. of this line must be

y(x) = 3x - 2.


Finding the gradient of a general function

Linear curves are simple, but how do we find the slope of any curve, y(x) at the point x ?


The gradient of the curve at point A is the same as that of the tangent at point A. So, all we need to do is construct the tangent and measure its gradient, Δy / Δx.

Example What is the gradient of y(x) = x2 - 4x - 1 when x = 4?

Solution Plot out the curve, then construct the tangent when x = 4 by eye, as best you can. Measure the gradient Δy / Δx by completing the triangle.


Graphically, we find that = 4.

Analytical Differentiation

Drawing tangents is a rather cumbersome method of obtaining gradients. Is there an analytic method?

The answer is differentiation. A simplified derivation of this is given in the handout, but we only really need to learn the 'magic formula' (see below).

Notation: The slope, or gradient, or differential, or derivative can be written in many equivalent ways:

y = = =

For other variable names and functions, there is the equivalent notation.

e.g. for s(t), we have ,

for E(ν), we have

for φ(λ), we have

Differentiation 'magic formula' (for standard polynomials)



To differentiate a polynomial function, multiply together the leading factor, a, and the exponent (power),. n, then subtract one from the exponent.

Examples

1. y = x2 , = 2x

2. y = 2x3, = 6x

3. y = 9x27, = 243x26

4. u = 3m6, = 18m5

5. φ = 7λ, = 7

6. Ψ= ,

7. p = -5q2, = -10q

8. y = 5, = 0       The differential of a constant is always zero, i.e. its slope is zero, as we'd expect.


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