5: The Chain Rule © Christine Crisp “Teach A Level Maths” Vol. 2: A2 Core Modules

The Chain Rule Module C3 "Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages"

The Chain Rule  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 rules we have developed for differentiating are: A reminder of the differentiation done so far!

The Chain Rule 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,. ) Suppose and Let Differentiating a function of a function.

The Chain Rule Consider again and. Let Differentiating both these expressions: Then, We must get the letters right. Let 23 )4(  x

The Chain Rule Consider again and. Let Differentiating both these expressions: Then, Let 23 )4(  x Now we can substitute for u

The Chain Rule Consider again and. Let Differentiating both these expressions: Then, Let 23 )4(  x Can you see how to get to the answer which we know is We need to multiply by

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

The Chain Rule So, we have So, and to get we need to multiply by 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.

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

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

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

The Chain Rule e.g. 2 Find if Solution: Whichever way we start we get and

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

The Chain Rule Exercise Use the chain rule to find for the following: Solutions: 1.

The Chain Rule Solutions 2. 3.

The Chain Rule Solutions 4.

The Chain Rule Solutions 5.

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

The Chain Rule Exercise Use the chain rule to differentiate the following: Solutions: 1.

The Chain Rule Solutions 2. 3.

The Chain Rule 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 has been multiplied by the derivative of the outer function The derivative of the inner 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. ) which is

The Chain Rule So, the chain rule says  differentiate the inner function  multiply by the derivative of the outer function e.g.

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

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

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

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

The Chain Rule 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.

The Chain Rule

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

The Chain Rule e.g. 1 Find if Solution: Let Then Differentiating: We need to recognise the function as and identify the inner function ( which is u ). We don’t multiply out the brackets

The Chain Rule The chain rule can also be used to differentiate functions involving e. Let e.g. Differentiate Solution: The inner function is the 1 st operation on x so here it is  2x.

The Chain Rule 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 has been multiplied by the derivative of the outer function The derivative of the inner function which is ( I’ve put dashes here because we want to ignore the inner function at this stage. We mustn’t differentiate it again. ) which is

The Chain Rule So, the chain rule says  differentiate the inner function  multiply by the derivative of the outer function ( The outer function is ) ( The inner function is ) e.g. With exponential functions, the index never changes.