A general approach to rates of change

Using a similar approach to that in the previous slide show it is possible to find the gradient at any point (x, y) on the curve y = f(x). At this point it is useful to introduce some “new” notation due to Leibniz. The Greek letter ∆(delta) is used as an abbreviation for “the increase in”. Thus the “increase in x” is written as ∆x, and the “increase in y” is written as ∆y. P (x 1, y 1 ) Q (x 2, y 2 ) x y O So when considering the gradient of a straight line, ∆x is the same as x 2 – x 1 and ∆y is the same as y 2 – y 1. ∆x∆x ∆y∆y Gottfried Leibniz gradient Note ∆x is the same as h used on the previous slide show.

Gradient of the curve y = x 2 at the point P( x, y ) Suppose that the point Q(x + ∆x, y + ∆y) is very close to the point P on the curve. The small change from P in the value of x is ∆x and the corresponding small change in the value of y is ∆y. It is important to understand that ∆x is read as “delta x” and is a single symbol. The gradient of the chord PQ is: x O y P(x, y) Q(x + ∆x, y + ∆y) y = x 2 x y y ∆x∆x ∆y∆y The coordinates of P can also be written as (x, x 2 ) and the coordinates of Q as [(x + ∆x), (x + ∆x) 2 ]. So the gradient of the chord PQ can be written as: = 2x + ∆x

So = 2x + ∆x As ∆x gets smaller approaches a limit and we start to refer to it in theoretical terms. This limit is the gradient of the tangent at P which is the gradient of the curve at P. It is called the rate of change of y with respect to x at the point P. For the curve y = x 2, = 2x This is the result we obtained previously. This is denoted by or.

Gradient of the curve y = f( x ) at the point P( x, y ) x O y P(x, y) Q(x + δx, y + δy) y = f(x) x y y δx δy For any function y = f(x) the gradient of the chord PQ is: The coordinates of P can also be written as (x, f(x)) and the coordinates of Q as [(x + δx), f(x + δx)]. So the gradient of the chord PQ can be written as:

It is defined by In words we say: The symbol is called the derivative or the differential coefficient of y with respect to x. “dee y by dee x is the limit of as δx tends to zero” “tends to” is another way of saying “approaches” DEFINITION OF THE DERIVATIVE OF A FUNCTION

Sometimes we write h instead of ∆x and so the derivative of f(x) can be written as DEFINITION OF THE DERIVATIVE OF A FUNCTION

If y = f(x) we can also use the notation: = f’ (x) In this case f’ is often called the derived function of f. This is also called f-prime. The procedure used to find from y is called differentiating y with respect to x.

Find for the function y = x 3. Example (1) In this case, f(x) = x 3. = 3x 2 f(x)f ‘ (x)f ‘ (x) x2x2 2x2x x3x3 3x23x2 Results so far

y = x 3

Gradient at (2, 8) = 12

Find for the function y = x 4. Example (2) In this case, f(x) = x 4. = 4x 3 Results so far f(x)f ‘(x)f ‘(x) x2x2 2x2x x3x3 3x23x2 x4x4 4x34x3

Find for the function y =. Example (3) In this case, f(x) =.

So we now have the following results: We also know that if y = x = x 1 then and if y = 1 = x 0 then These results suggest that if y = x n then It can be proven that this statement is true for all values of n. nxnxn –1 2x2x2 2x2x 3x3x3 3x23x2 4x4x4 4x34x3