In general, it is not necessary to solve an equation of *y *in terms of *x*in order to differentiate the functions defined by the equation. To illustrate this, consider the equation

* xy =* 1

A way to find *dy*/*dx *is to rewrite this equation as

of which one has to

However, there is another way to obtain this derivative. We can differentiate both sides of *xy =* 1 before settling for * y* in terms of *x*treating* y* (temporarily unspecified) a differentiable function of *x*. With this approach, we get

If we now replace in the last expression we get

which is in accordance with . This method to get derivatives is called *implicit differentiation.*

Use implicit differentiation to find *dy*/*dx* if

Solving for *dy*/*dx *we get

Note that this formula involves both *x* and* y*. In order to get a formula for *dy*/*dx *that only involves *x*we would have to solve the original equation to * y* in terms of * x * and then replace in . However, this is impossible to do; thus we are forced to leave the formula *dy*/*dx* in terms of *x* and *y.*

Use implicit differentiation to find if .

** Solution.** Differentiated both sides of implicitly, one obtains

that we get

Differentiating both sides of implicitly, one obtains

Replacing inside of and simply put, using the original equation, we get

We** Examples 1 and 2**, the results of the formulas for *dy*/*dx *involve both * x* and *y*. Although it is usually more desirable to have the formula for *dy*/*dx *expressed only in terms of *x*, have the formula in terms of* x* and* y* is not an impediment to finding the inclinations and equations of the tangent lines as long as the coordinates * x* and * y* of the tangency point are known.