3.6 - Implicit Differentiation (1)

# 3.6 - Implicit Differentiation (1) - Your name Implicit...

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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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3.6 - Implicit Differentiation (1) - Your name Implicit...

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