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Computing Derivatives Using Implicit Differentiation

We've taken lots of derivatives of explicit functions. In fact, so far we've only taken derivatives of explicit functions. We've had an equation for y (or f, or whatever) and we've used our collection of rules to find f '.

We can also take derivatives of implicit functions. If we have an equation relating x and y, we can take derivatives first and solve for y' later.

We can use either Leibniz or Lagrange notation.

It will be helpful to have a variation on Lagrange notation. The expression (...) means, "take the derivative, with respect to x, of whatever is in the parentheses." It essentially means the same thing as (...)', but with the added precise statement that yes, we are taking the derivative with respect to x and not with respect to anything else.

For example,

and

Remember, means the same thing as a prime.

(f + g)' = f ' + g '

We can also say