1. Differentiation Summary

2. 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”

3. e.g.

4. 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

5. 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

6. 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

7. 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

8. 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.

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

10. 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

11. 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.

12. 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.

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

14. 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.

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

16. 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

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