3.6 implicit differentiation and rational exponents
TRANSCRIPT
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3.6 Implicit DifferentiationAnd Rational Exponents
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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:
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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.
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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.
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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”
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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
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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
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Examples
6. Find the slope at (-1, 1) for 2 2x y 2