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

Video Transcript

Find 𝑑𝑦 𝑑𝑥, given that 𝑦 equals negative 43 over 𝑥 to the power of eight.

So to find 𝑑𝑦 𝑑𝑥, what we’re gonna need to do is actually differentiate our function. So the first thing I’m gonna do is I’m actually going to rewrite our function. And in order to do that, what I’m gonna use is an exponent rule. And the exponent rule is that one over 𝑎 to the power of 𝑏 is equal to 𝑎 to the power of negative 𝑏. So therefore, we can say that 𝑦 is equal to negative 43𝑥 to the power of negative eight. And I’ve done this so that we can actually remove the fraction and it makes it easier to differentiate.

Okay, now, we’re actually gonna move on to the differentiation. And if we want to differentiate, we think about our function in the form 𝑎𝑥 to the power of 𝑏. So if we have a function in this form, then we can say that the derivative which I’ve denoted here is 𝑑 𝑑𝑥 of 𝑓 𝑥 is gonna be equal to 𝑎𝑏 𝑥 to the power of 𝑏 minus one. So what I’ve done here is I’ve actually multiplied our coefficient by our exponents — so 𝑎 and 𝑏. And then, we’ve reduced the exponent by one. So we’ve got 𝑏 minus one.

Okay, great, we know what to do. It’s to use this to actually differentiate our function. So then, we can say that the derivative of our function or 𝑑𝑦 𝑑𝑥 is equal to negative 43 multiplied by negative eight cause that’s our coefficient multiplied by our exponent and then 𝑥 to the power of negative eight minus one. Right, so now, we can actually simplify this and this gives us 344𝑥 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 we’ve got 𝑥 to the power of negative nine because negative eight minus one gives us negative nine.

So fantastic, we’ve actually reached an answer here. We’ve differentiated our term. The final thing we’re gonna do is actually rewrite it in our original form — so include a fraction again. And in order to do this, what I’ll use is the exponent rule that I used earlier just in the opposite way. And that rule was the one over 𝑎 to the power of 𝑏 equals 𝑎 to the power of negative 𝑏. So we can say that given that 𝑦 equals negative 43 over 𝑥 to the power of eight 𝑑𝑦 𝑑𝑥 is equal to 344 over 𝑥 to the power of nine.