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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
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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
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HCF is 5
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Key differentiating facts
Chain Rule
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Therefore:
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Product Rule
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We can now locate the turning points where:
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This cannot currently be simplified, though you will meet how
to in Core 4 |
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Quotient Rule
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Remember:
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So turning point where numerator equals zero
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Implicit differentiation
When you differentiate y with respect to x
you must remember to multiply by 
. This basically comes from the chain rule.
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Differential with respect to x |
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Examples of implicit differentiation
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Differentiate each term in turn Re-arrange to find in terms of |
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Use the product rule for the left hand side.
Differentiate |
Applications of the chain rule
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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:
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Rate of change of radius
We need to find We know that:
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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,
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Rate of change of radius
We need to find We know that:
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