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Core 3 revision guides
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. Rewrite 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 Rearrange 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 