Page 26 - Calc for the Clueless

Only the chain rule is necessary.

Example B—

Do not use quotient rule.

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Rewrite example as y =7x and simply use power rule.

Example C—

-8
2
Rewrite as y = b(x + 5) .

Implicit Differentiation

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3
3
Suppose we are given y + x y + X =9. It would be difficult, maybe impossible, to solve for y. However, there
is a theorem called the implicit function theorem that gives conditions that will show that y = f(x) exists even if
we can never find y. Moreover, it will allow us to find dy/dx even if we can never find y. Pretty amazing, isn't
it?!!!!!
Let f(y) = y where y = y(x). Using the chain rule and power rule, we get
n

Example 16—

Find dy/dx if y + x y + x = 9.
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3
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1. We will differentiate straight across implicitly.

2. We will differentiate the first term implicitly, the second term implicitly and with the product rule, and the
rest the old way.

3. We will solve for dy/dx using an algebraic trick that can save up to five algebraic steps. With a little practice,
you can save a lot of time!!!!

Let's do the problem.