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Unformatted text preview: Your name Implicit Differentiation Section 3.6 Why do we need implicit differentiation? VAll of the functions you have seen so far in
this class could easily be written explicitly.
whenever this is the case, you don't need to
use implicit differentiation. But what do you
do if it is very difficult or impossible to isolate
y? This is when we need implicit
differentiation. “Example: Solva ny + yzx a #2 for y. Guidelines for implicit Differentiation / Differentiate both sides of the equation with
respect to x. r Collect all terms involving dy/dxon the left
side of the equation and move all other terms
to the right side of the equation. r Factor dy/dxout of the left side of the
equation. « Isolate dy/dx by dividing. Implicit Differentiation implicit vs. Explicit /y is written explicitly as a function of x
when y is isolated on one side of the
equation. wWhen y is not isolated we say the
function is written implicitly. Keep in Mind «When ﬁnding dy/dx and you
are differentiating a term
involving xalone, you can
differentiate as usual. « When you differentiate terms
involving y, you must apply the
Chain Rule. because you are
assuming that yis defined
implicitly as a function of x. Examples
x3 + y“ = 7y — 5
xzy + yzx = —2
2 sin xcos 3/: I Find the normal line to the curve
y=25in(1r x — y) at the point (i .0) 9/23/2009 Your name 9/23/2009 Finding a 2nd Derivative Implicitly Examples I y/ xas e are Find oer/0')?? « Differentiate both sides to get the 2nd
derivative. You will have to use implicit « xy + y? = l
differentiation on the right—hand side. « Substitute your answer to the first step for
dy/dx. « If possible. substitute in the original equation
to simplify your answer. Implicit Differentiation 2 ...
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 Spring '11
 PamelaSatterfield
 Implicit Differentiation

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