Implicit Differentiation (Optional)

Implicit differentiation is used when \(x\) and \(y\) are mixed together in an equation and it’s difficult (or impossible) to make \(y\) the subject.
Key Idea:
Differentiate both sides of the equation with respect to \(x\), treating \(y\) as a function of \(x\) and applying the chain rule to include the derivative of \(y\), \(\frac{\mathrm{d}y}{\mathrm{d}x}\). Use the product rule of differentiation, where necessary.
For example:
Given a circle with equation \(\begin{matrix} {{x}^{2}}+{{y}^{2}}=25 \\ \end{matrix}\), find an expression for \(\frac{\mathrm{d}y}{\mathrm{d}x}\) in terms of \(x\) and \(y\).