To solve this problem I will use 3 techniques:
Implicit differentiation. This involves differentiation of both sides of the equation.
The product rule :
The chain rule. This involves differentiation of the outer layer and multiplying by the derivative of the inner layer.
I will show how this applies to the 1st derivative then work through the 2nd.
Using rule 1 we can write:
I'll differentiate each term:
Factorising we get:
Now we differentiate again using the same techniques:
This simplifies to:
The original equation is
We can put these values back into Eqn 1: