Differentiation in Core 3 (C3) are an extension of the work that you did in Core 1 and Core 2.
For the C3 module, there are a few standard results for differentiation that need to be learnt. These are:
The Chain Rule is used to differentiate when one function is applied to another function. A typical example of this is:
One of the ways of remembering the chain rule is: Find the derivative outside, then multiply it by the derivative inside. In the example above, this becomes:
The product rule is used when two functions are multiplied together.
The quotient rule is used when one function is divided by another. It is a specific case of the product rule. A typical example of this is:
Implicit differentiation is used when a function is not a simple but contains a mixture of x and y parts. A typical example of this is to differentiate:
When differentiating the y components of the expression you differentiate as normal, and then multiply by . So differentiating both sides of the above expression it becomes:
Then by factorising the left hand side and cancelling, this becomes: