Differentiation Summary

SUMMARY The gradient at a point on a curve is defined as the gradient of the tangent at that point The process of finding the gradient function is called differentiating The function that gives the gradient of a curve at any point is called the gradient function The rule for differentiating terms of the form is “power to the front and multiply” “subtract 1 from the power”

e.g.

e.g. Find the gradient at the point where x = 1 on the curve Solution: Differentiating to find the gradient function: When x = 1, gradient m = Using Gradient Functions

Exercises Find the gradients at the given points on the following curves: 1. at the point When 2. at the point When 3. at the point When

SUMMARY To find the point(s) on a curve with a given gradient: find the gradient function let equal the given gradient solve the resulting equation

Points with a Given Gradient e.g. Find the coordinates of the points on the curve where the gradient equals 4 Gradient of curve = gradient of tangent = 4 We need to be able to find these points using algebra

The stationary points of a curve are the points where the gradient is zero A local maximum A local minimum x The word local is usually omitted and the points called maximum and minimum points. e.g.

e.g.1 Find the coordinates of the stationary points on the curve Solution: or The stationary points are (3, -27) and ( -1, 5)

For a max we have The opposite is true for a minimum At the max On the right of the max On the left of the max Calculating the gradients on the left and right of a stationary point tells us whether the point is a max or a min. Determining the nature of a Stationary Point

At the max of the gradient is 0, but the gradient of the gradient is negative. The gradient function is given by e.g. Consider Another method for determining the nature of a stationary point.

The notation for the gradient of the gradient is “d 2 y by d x squared” At the min of The gradient function is given by the gradient of the gradient is positive.

The gradient of the gradient is called the 2nd derivative and is written as

e.g. Find the stationary points on the curve and distinguish between the max and the min. Solution: Stationary points: or We now need to find the y-coordinates of the st. pts.

max at min at At , To distinguish between max and min we use the 2nd derivative,

SUMMARY To find stationary points, solve the equation maximum minimum Determine the nature of the stationary points either by finding the gradients on the left and right of the stationary points or by finding the value of the 2nd derivative at the stationary points

SUMMARY COULD ALSO BE A POINT OF INFLECTION! gradients on the left and right of the stationary points