1. Ch.3 The Derivative

2. Differentiation Given a curve y = f(x ) Want to compute the slope of tangent at some value x=a. Let A=(a, f(a )) and B=(a +  x, f(a +  x )) = (a +  x, f(a) +  y) where the change in y -value (  y) is given by  y = f(a +  x ) – f(a) and B is a point on the curve y = f(x ) close to A.

3. Therefore slope of chord

4. Definition The derivative of f is defined by This gives the slope of tangent at x = a after setting x =a +  x. We say that f is differentiable at a if this limit exists.

5. Thus the derivative of f at x is Just set h =  x and a=x. We say that f is differentiable at x if the above limit exists. This defines a new function f’(x) called the derivative of f(x).

6. Other notations for f’(x ) Examples 1., show that

8. 2.

9. 3.

10. Trigonometry identity: Example

12. Example: Is f(x )=| x | differentiable at x=0 ? Must consider So the limit does not exists that is y =| x | is not differentiable at 0

13. Thus continuity does not imply differentiability.

14. Rate of Change Recall that the slope of the tangent at a point measures the (instantaneous) rate of change of y with respect to x at that point. Example. Let s(t) be the distance of a car that has traveled at time t. Speed v= rate of change of distance

15. Similarly acceleration a = rate of change of speed

16. Example A cylindrical tank holds 50 litres of water and can be drained from the bottom of the tank in 100 seconds. Find the rate of change of volume after 30 seconds given volume V of water in the tank after t seconds can be shown to be Rate of change of volume

17. Theorem If f(x ) is differentiable at a then f is continuous at x=a. Proof. Assume f(x ) differentiable at x=a. Must show But

18. Basic Differentiation Rules Read Section 3.2 (or the same topic from other textbooks) You should be able to use the differentiation rules/theorems to find the derivatives of functions