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Implicit Differentiation. 2.5a. Implicit Differentiation. So far, all the equations and functions we looked at were all stated explicitly in terms of one variable:.

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implicit differentiation1
Implicit Differentiation

So far, all the equations and functions we looked at were all stated explicitly in terms of one variable:

In this function, y is defined explicitly in terms of x. If we re-wrote it as xy = 1, y is now defined implicitly in terms of x.

It is easy to find the derivative of an explicit function, but what about:

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This is not a function, but it would still be nice to be able to find the slope.

Do the same thing to both sides.

Note use of chain rule.

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1 Differentiate both sides w.r.t. x.

2 Solve for .

This can’t be solved for y.

This technique is called implicit differentiation.