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Find the derivative of each function by using the quotient rule.$$P=\frac{8 i^{2}}{4-3 i}$$

$\frac{d P}{d i}=\frac{64 i-24 i^{2}}{(4-3 i)^{2}}$

Calculus 1 / AB

Chapter 23

The Derivative

Section 6

Derivatives of Products and Quotients of Functions

Derivatives

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we are given P is equal to eight I squared. And I'm assuming that this i is the independent variable, not the imaginary unit. So if it's supposed to be the imaginary unit, uh then I might be doing the saints you're wrong. But the first thing to recognize is if we really are doing the quotient rule, uh identify your question is the division. So when I do the derivative of P with respect to I you take the derivative of the top. So that will be 69. Following our power rule. You leave the bottom alone and then minus now you take the derivative of the bottom, which is negative three. And you leave the top alone all over the denominator for minus three I squared. So as I go to simplify this, because a lot of teachers would actually make you simplify this, you can distribute and 16 times four is 64 I uh and then 16 times three would be 48 I squared. And then over here we would have a plus 20 for uh squeeze that right there I squared. Uh So if I were asking my students to simplify this, um I would hope that they would combine my terms. So in the numerator we would have 64 I minus 24 I squared. Now you might be able to simplify that and cancel. Um but most times uh this is the best answer you can come up with. Um But I do want to point out because some people might factor out of four or an eight in the numerator. If you did that, you would have um and an eye for the matter eight minus three. I and you can see that nothing cancels out, so I wouldn't do this, but maybe your teacher wants you to, I would stop right here.

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