3.6 Implicit DifferentiationAnd Rational Exponents

Rational Exponents are easy!

These problems will have exponents that aren’t integers; they will most likely be fractions.

Hmmm… how could an exponent be a fraction??

That’s right! A radical!!

It doesn’t matter. You will still use the generalized power rule:

Examples

1. Find if

2. Find if

dydx

3 2y 12x 6

dydx

3 2 2(x 1)y

x

Functions do NOT need to be rationalized; only numbers need to be rationalized.

Implicit Differentiation

This process is used when y cannot be isolated. We will look at a problem where y CAN be isolated to understand the idea, but in many problems y will be mixed in as a product or a quotient.

We will be treating y as a differentiable functions.

Let’s try this one

y x 2y x

12y x

12dy 1

xdx 2

dy 1dx 2 x

dy

2y 1dx

dy 1dx 2y

Same thing

right?

The derivative of y in terms of x has to have that extra symbol because “y” is not “x”

What does this mean?

WHENEVER you have to take a derivative of

y, tack on . No kidding.

Then, isolate . There might be y in your

answer; it also might be easy to sub back in y.

Lets see an example… or two….

dydx

dydx

Examples

3. Find if

4. Find if

5. Find if (hint: continue #4)

dydx

2 2xy x y 4

dydx

x y 5

Leave y in the answer.

2

2

d ydx

x y 5

Examples

6. Find the slope at (-1, 1) for 2 2x y 2