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Problem 48 Medium Difficulty

The Power Rule can be proved using implicit differentiation for the case where $ n $ in a rational number, $ n = p/q, $ and $ y = f(x) = x" $ is assumed beforehand to be a differentiable.
function. If $ y = x^{p/q}, $ then $ y^q = x^p. $ Use implicit differentiation to show that
$ y' = \frac {p}{q} x^{(p/q)-1} $

Answer

$$
y^{\prime}=\left(\frac{p}{q}\right) \cdot x^{(p / q)-1}
$$

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Video Transcript

in this problem Where? Excuse him. Said the appreciation to prove So we're gonna use this or less. Take debatable bullet size. We dissect X. We have two hands Y times humans for 10 swipe crimes. This wise about your X is equal to p times X to the C minus one. From this we see that then why Prime is off for p times extent P minus one B y a que times what humanise form So we can write this one as peor que times x to the p minus one divided by a white B Q minus one. Now we know why in terms of X and that is given right here. So we're just gonna plug down and we have then pure ju plans extra the P minus one divided by X to the A P or two the power all to minus one that is that equal to pure Q times X to the power P minus one minus a few times too, plus p you are, aren't you? We can let this one has your work. You terms eggs to the P to brightness Jew minus B June plus p invited ju bees will go away. Um, we then will have, um peor que terms x to the A P miners start peor que minus one, and we actually can write this one. That's pure Q times X to the P over few minus one. And we just proved whatever it is, give any problems.