# Untitled Page 24

- Page ID
- 148365

## 5.2 Implicit differentiation

We can differentiate any function that is written as a formula,
and find a result in terms of a formula. However, sometimes the
original problem can't be written in any nice way as a formula.
For example, suppose we want to find *dy*/*dx* in a case
where the relationship between *x*
and *y* is given by the following equation:

*y*

^{7}+

*y*=

*x*

^{7}+

*x*

^{2}.

There is no equivalent of the quadratic formula for seventh-order
polynomials, so we have no way to solve for one variable in terms
of the other in order to differentiate it. However, we can still
find *dy*/*dx* in terms of *x* and *y*. Suppose we let *x*
grow to *x*+*dx*. Then for example the *x*^{2} term will grow
to (*x*+*dx*)^{2}=*x*+2*dx*+*dx*^{2}. The squared infinitesimal
is negligible, so the increase in *x*^{2} was really just
2*dx*, and we've really just computed the derivative of
*x*^{2} with respect to *x* and multiplied it by *dx*. In
symbols,

That is, the change in *x*^{2} is 2*x* times the change in *x*.
Doing this to both sides of the original equation, we have

This still doesn't give us a formula for the derivative in
terms of *x* alone, but it's not entirely useless. For instance,
if we're given a numerical value of *x*, we can always use
Newton's method to find *y*, and then
evaluate the derivative.