HOW TO DO IT
By now, it should be easy for you to take the derivative of an equation, such as y = 3x^5 − 7x. If you’re
given an equation such as y^2 = 3x^5 − 7x, you can still figure out the derivative by taking the square root of
both sides, which gives you y in terms of x. This is known as finding the derivative explicitly. It’s messy,
but possible.
If you have to find the derivative of y^2 + y = 3x^5 − 7x, you don’t have an easy way to get y in terms of x, so
you can’t differentiate this equation using any of the techniques you’ve learned so far. That’s because each
of those previous techniques needs to be used on an equation in which y is in terms of x. When you can’t
isolate y in terms of x (or if isolating y makes taking the derivative a nightmare), it’s time to take the
derivative implicitly.
Implicit differentiation is one of the simpler techniques you need to learn to do in calculus, but for some
reason it gives many students trouble. Suppose you have the equation y^2 = 3x^5 − 7x. This means that the
value of y is a function of the value of x. When we take the derivative, , we’re looking at the rate at
which y changes as x changes. Thus, given y = x^2 + x, when we write
= 2x + 1we’re saying that “the rate” at which y changes, with respect to how x changes, is 2x + 1.
Now, suppose you want to find . As you might imagine
So here, . But notice that this derivative is in terms of x, not y, and you need to find the
derivative with respect to y. This derivative is an implicit one. When you can’t isolate the variables of an
equation, you often end up with a derivative that is in terms of both variables.
Another way to think of this is that there is a hidden term in the derivative, , and when we take the
derivative, what we really get is