Question Video: Finding the First Derivative of a Function Involving Negative Exponents Using the Power Rule | Nagwa Question Video: Finding the First Derivative of a Function Involving Negative Exponents Using the Power Rule | Nagwa

Question Video: Finding the First Derivative of a Function Involving Negative Exponents Using the Power Rule Mathematics • Second Year of Secondary School

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Find ππ¦/ππ₯, given that π¦ = β43/π₯βΈ.

02:19

Video Transcript

Find ππ¦ ππ₯, given that π¦ equals negative 43 over π₯ to the power of eight.

So to find ππ¦ ππ₯, what weβre gonna need to do is actually differentiate our function. So the first thing Iβm gonna do is Iβm actually going to rewrite our function. And in order to do that, what Iβm gonna use is an exponent rule. And the exponent rule is that one over π to the power of π is equal to π to the power of negative π. So therefore, we can say that π¦ is equal to negative 43π₯ to the power of negative eight. And Iβve done this so that we can actually remove the fraction and it makes it easier to differentiate.

Okay, now, weβre actually gonna move on to the differentiation. And if we want to differentiate, we think about our function in the form ππ₯ to the power of π. So if we have a function in this form, then we can say that the derivative which Iβve denoted here is π ππ₯ of π π₯ is gonna be equal to ππ π₯ to the power of π minus one. So what Iβve done here is Iβve actually multiplied our coefficient by our exponents β so π and π. And then, weβve reduced the exponent by one. So weβve got π minus one.

Okay, great, we know what to do. Itβs to use this to actually differentiate our function. So then, we can say that the derivative of our function or ππ¦ ππ₯ is equal to negative 43 multiplied by negative eight cause thatβs our coefficient multiplied by our exponent and then π₯ to the power of negative eight minus one. Right, so now, we can actually simplify this and this gives us 344π₯ to the power of negative nine. And we got 344 because we got negative 43 multiplied by negative eight. And a negative multiplied by a negative gives us a positive. And weβve got π₯ to the power of negative nine because negative eight minus one gives us negative nine.

So fantastic, weβve actually reached an answer here. Weβve differentiated our term. The final thing weβre gonna do is actually rewrite it in our original form β so include a fraction again. And in order to do this, what Iβll use is the exponent rule that I used earlier just in the opposite way. And that rule was the one over π to the power of π equals π to the power of negative π. So we can say that given that π¦ equals negative 43 over π₯ to the power of eight ππ¦ ππ₯ is equal to 344 over π₯ to the power of nine.

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