Question

In your own words, state the guidelines for implicit differentiation.

Short answer

Implicit differentiation involves: identifying the implicit function, differentiating with respect to x while treating y as a function of x and using the chain rule for terms involving y, and finally solving for \(y'\) or \(\frac{dy}{dx}\).

Step-by-step solution

Specify Function

The first step to understanding implicit differentiation is to specify a function. Typically in explicit functions, it is clear to solve for \(y\) in terms of \(x\). But, in implicit functions, \(y\) is not solved in terms of \(x\). That is, \(y\) is an implicit function of \(x\). For example, consider the implicit function \(x^2 + y^2 = 9\).

Differentiate the Function with Respect to x

Despite the function being in terms of \(x\) and \(y\), the differentiation is with respect to \(x\). Hence, the power rule applies to the \(y\) variable as well, but you must add on \(y'\) (or \(\frac{dy}{dx}\)) at the end of every term involving \(y\). This is because of the chain rule - you're differentiating the outer function \(f(y)\), and then multiplying by the derivative of the inner function \(y(x)\). In our example, differentiating both sides of the equation with respect to \(x\) gives: \(2x + 2yy' = 0\).

Solve for \(y'\) (or \(\frac{dy}{dx}\))

After differentiating, the next step is to solve for the derivative \(y'\) (or \(\frac{dy}{dx}\)). This would be your final answer. In the example, solving for \(y'\) would give \(y' = -\frac{x}{y}\).