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Core 3: Differentiation Skills

Contents

1)           Factorising algebraic expressions

2)          Key differentiating facts

3)          Chain Rule

4)          Product Rule

5)          Quotient Rule

6)          Implicit differentiation

7)          Applications of the chain rule


 

Factorising algebraic expressions

 

 

 

 

 

First look for the highest common factor.

In this case it is 1

Then, look for the comparable terms.

Find the lowest power of each comparable term.

Re-write the highest power of each comparable term with the lowest power as a base

 

 

 

 

 

 

HCF is 5

 

 

 


 

Key differentiating facts


 

Chain Rule

Therefore:

 

 

 

 


 

Product Rule

 

 

 

 

 

We can now locate the turning points where:

 

 

 

 

This cannot currently be simplified, though you will meet how to in Core 4

 


 

Quotient Rule

 

 

 

 

 

 

 

 

 

Remember:

 

 

 

 

 

 

 

So turning point where numerator equals zero

 

 

 

 

 


 

Implicit differentiation

When you differentiate y with respect to x you must remember to multiply by . This basically comes from the chain rule.

Function

Differential with respect to x

 

Examples of implicit differentiation

 

Differentiate each term in turn

 

Re-arrange to find in terms of

 

 

 

 

Use the product rule for the left hand side.

Differentiate as normal

 


 

Applications of the chain rule

A circle of liquid lies on a table.

The radius is increasing at a rate of 0.2 cm per second.

Find an expression for the rate at which the area is increasing

 

 

So, for example, when the radius=10:

 

Rate of change of radius

 

We need to find

 

We know that:

so

 

 

A spherical bubble has a volume which is decreasing at a rate of 0.7cm per second. Find an expression for the rate at which the radius is decreasing

 

 

So, for example, when the radius is 5,

 

Rate of change of radius

 

We need to find

 

We know that:

so