A Hader Implicit Differetiation Problem

by Eric Ndegwa
(Nairobi, Kenya)

How do we find the second derivative of an implicit function? The
example posted by Eric is the following:

Find the second derivative with respect to x of the function y = f(x) defined by:

In this second we're asked the second derivative of y with respect to x. That is nothing more than the derivative of the derivative. So, we must first find the derivative. We have a quotient, and as you know, I don't memorize the quotient rule, I just first write the function as a product:

And now we take the derivative on both sides:

And this will be messy, but now we now need to take the derivative of both sides again. In the second term we have tree factors in a product, so it would be clearer if we first make a little change of variables before we apply the product rule. We can make:

So, we have:

And now we derive:

And now, if you substitute back u' you'll get your horrible answer. This problem is really an exercise in patience and attention.

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