Implicit Differentiation

In general, it is not necessary to solve an equation of y in terms of xin 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 xtreating 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 xwe 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.

Next: Derivatives of Rational Powers of x