1. ex, lnx and the chain rule
Differentiating :
ex, lnx and the chain rule
Objectives
To understand why the differentials of lnx and ex are what they are!
To be able to differentiate more complex functions that are composite functions
© Christine Crisp

2. The Gradient at a point on a Curve
Definition: The gradient of a point on a curve equals the gradient of the tangent at that point.
e.g.
Tangent at
(2, 4) x 12 3
The gradient of the tangent at (2, 4) is
So, the gradient of the curve at (2, 4) is 4

3. The Rule for Differentiation
The gradient of this curve at any point is the tangent
The tangent changes at any one point for a curve

6. The gradient Dee-Y by Dee-X
The gradient for a function is defined as
Use this to show that the gradient of y=x2 is 2x
2 minutes for this !

7. Solution Like this....

8. Differentiate ex and ln x
From first principles
This means use the definition of gradient as a limit
1 minute to think about this

9. Let’s consider the derivative of the exponential function.
Going back to our limit definition of the derivative:
First rewrite the exponential
using exponent rules.
Next, factor out ex.
Since ex does not contain h, we
can move it outside the limit.

10. Substituting h=0 in the limit expression results in the
indeterminate form , thus we will need to determine it.
We can look at the graph of and observe what happens as x gets close to 0. We can also create a table of values close to either side of 0 and see what number we are closing in on.
X≠0 of course
Table
Graph x y
At x = 0, f(0) appears to be 1.
As x approaches 0, y approaches 1.

11. We can safely say that from the last slide that
Thus
Rule 1: Derivative of the Exponential Function
The derivative of the exponential function is the
exponential function.

12. What about ln x Method 1 – Using previous result
Method 2 – Consider Limits

13. What about ln x Method 1 – Using previous result y = ln x therefore
Method 2 – Consider Limits
y = ln x
therefore
ey = x ( raise both sides to e) dx/dy = ey. (Oh yes it does!!)
Since dx/dy * dy/dx = 1
1/(dx/dy) = dy/dx.
= 1/ey f'(x) = 1/x. (as ey=x)
f(x) = ln x
f(x+h) = ln (x+h)
f'(x) = lim(h->0) [ln (x+h) - ln x] / h
= lim(h->0) ln (x+h/x) / h
= lim(h->0) 1/h * ln(1+h/x) Since lim(h->0) ln(1+h/x) -> h/x where h ≠ 0,
f'(x) = 1/h * h/x = 1/x

15. Exercise

16. A reminder of the differentiation done so far!
The gradient at a point on a curve is defined as the gradient of the tangent at that point.
The function that gives the gradient of a curve at any point is called the gradient function.
The process of finding the gradient function is called differentiating.
The rules we have developed for differentiating are:

17. Differentiating a function of a function .
Suppose and
Let
We can find by multiplying out the brackets:
However, the chain rule will get us to the answer without needing to do this ( essential if we had, for example, )

18. ) 4 ( - x Consider again and . Let Let Then,2 3 ) 4 ( - x
Let
Then,
Differentiating both these expressions:
We must get the letters right.

19. ) 4 ( - x Consider again and . Let Let Then,2 3 ) 4 ( - x
Let
Then,
Differentiating both these expressions:
Now we can substitute for u

20. ) 4 ( - x Consider again and . Let Let Then,2 3 ) 4 ( - x
Let
Then,
Differentiating both these expressions:
Can you see how to get to the answer which we know is
We need to multiply by

21. So, we have
and to get we need to multiply by
So,
This expression is behaving like fractions with the ‘s on the r,h,s, cancelling.

22. So, we have
and to get we need to multiply by
So,
This expression is behaving like fractions with the s on the r,h,s, cancelling.
Although these are not fractions, they come from taking the limit of the gradient, which is a fraction.

23. e.g. 1 Find if
We need to recognise the function as and identify the inner function ( which is u ).
Solution:

24. e.g. 1 Find if
We need to recognise the function as and identify the inner function ( which is u ).
Solution:
Then
Let
Differentiating:
Substitute for u
Tidy up by writing the constant first
We don’t multiply out the brackets

25. e.g. 2 Find if
Solution: We can start in 2 ways. Can you spot them?
Either write
and then let
Always use fractions for indices, not decimals. so
Or, if you don’t notice this, start with
Then

26. e.g. 2 Find if
Solution: Whichever way we start we get
and

27. SUMMARY
The chain rule is used for differentiating functions of a function. If
where , the inner function.

28. Exercise
Use the chain rule to find for the following: 1. 2. 3. 4. 5. 1.
Solutions:

29. Solutions 2. 3.

30. Solutions 4.

31. Solutions 5.

32. The chain rule can also be used to differentiate functions involving e.
e.g. 3. Differentiate
Solution: The inner function is the 1st operation on x, so u = -2x.
Let

33. Exercise
Use the chain rule to differentiate the following: 1. 2. 3.
Solutions: 1.

34. Solutions 2. 3.

35. Later we will want to reverse the chain rule to integrate some functions of a function.
To prepare for this, we need to be able to use the chain rule without writing out all the steps.
e.g. For we know that
The derivative of the inner function which is
has been multiplied by the derivative of the outer function
which is
( I’ve put dashes here because we want to ignore the inner function at this stage. We must not differentiate it again. )

36. So, the chain rule says
differentiate the inner function
multiply by the derivative of the outer function
e.g.

37. So, the chain rule says
differentiate the inner function
multiply by the derivative of the outer function
e.g.
( The inner function is )

38. So, the chain rule says
differentiate the inner function
multiply by the derivative of the outer function
e.g.
( The inner function is )
( The outer function is )

39. So, the chain rule says
differentiate the inner function
multiply by the derivative of the outer function
e.g.
( The inner function is )
( The outer function is )

40. Below are the exercises you have already done using the chain rule with exponential functions.
See if you can get the answers directly. 1. 2. 3.
Answers: 1. 2. 3.
Notice how the indices never change.

41. TIP: When you are practising the chain rule, try to write down the answer before writing out the rule in full. With some functions you will find you can do this easily.
However, be very careful. With some functions it’s easy to make a mistake, so in an exam don’t take chances. It’s probably worth writing out the rule.

42. The chain rule is used for differentiating functions of a function.
where , the inner function. If
SUMMARY