Implicit Differentiation

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So far we have differentiated functions in the form

so that y is expressed as a function of x. This is called explicit differentiation. We now want to look at differentiating expressions such as

where y is not given explicitly. This is called implicit differentiation.

To differentiate the above equation, w.r.t. x, you just differentiate each term separately. This is straightforward for the x² term and the 4, but how do you differentiate y²w.r.t x?

In fact you use the chain rule. We can say that

Then applying the chain rule

we get

and we can work out that

so we have

We can now differentiate our original equation

We could in this case, have re-arranged to make y the subject, then differentiated, but this would have been much more complicated, and in any case, there will be occasions when it is not possible to make y the subject.

Let's consider a second example- Differentiate, w.r.t. x

Using the chain rule
so

The general rule for differentiating a y term w.r.t. x is, Diff the term w.r.t. y and multiply by dy/dx.

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