In this section we learn how to perform implicit differentiation.
This technique will allow us to differentiate expressions in which \(y\) is not explicitly defined as a function of \(x\).
For instance, expressions like:
\[3 y^2x - sin(y)+4x^2=3\]
The method, or technique for differentiating expressions in which \(y\) is implicitly defined is shown here.
To differentiate functions in which \(y\) is implicitly defined, two things should be kept in mind:
Differentiate the expression that follows with respect to \(x\): \[3y^2-x^2=8\]
At times we'll be required to find an expression for the second derivative \(\frac{d^2y}{dx^2}\) (or higher).
To do this we "simply" differentiate the expression, with respect to \(x\), twice and rearrange to make \(\frac{d^2y}{dx^2}\) the subject.
Given the expression: \[3x^2+y^2 = 7\]