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