# Question Video: Differentiating a Combination of Functions Involving Exponential Functions Using the Product Rule Mathematics • Higher Education

Determine the derivative of π¦ = π^(β5π₯) π₯Β².

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### Video Transcript

Determine the derivative of π¦ equals π to the power of negative five π₯ multiplied by π₯ squared.

Now to enable us to actually determine the derivative, what weβre gonna use is the product rule. And we can actually use the product rule when we have our function in the form π¦ equals π’π£. And if we take a look at our function, itβs actually in this form. So how does the actual product rule work? Well the product rule tells us that the derivative is equal to π’ ππ£ ππ₯ plus π£ ππ’ ππ₯.

So what this means is π’ multiplied by the derivative of π£ plus π£ multiplied by the derivative of π’. Okay, great! So now that we know the product rule, letβs use it to actually determine our derivative. So in order to actually determine our derivative, first of all what we need to do is to decide what π’ and π£ are. So π’ is gonna be equal to π to the power of negative five π₯, and π£ is gonna be equal to π₯ squared.

Next, what we want to do is actually differentiate our π’ and π£. So Iβm gonna start with π£ because this is more straightforward. So we can say that dπ£ dπ₯ is going to be equal to the derivative of π₯ squared. Well this is gonna be equal to two π₯, just remind us how we did that. So our exponent multiplied by our coefficient, so two multiplied by one, and then itβs π₯ to the power of, and then we reduce our exponent by one, so two minus one, which just be one. So we get two π₯.

So now we can move on to π’. So if we wanna find dπ’ dπ₯, well weβre actually gonna have to use is a general rule to help us here as well. And thatβs because π’ is in the form π¦ is equal to π to the power of π of π₯. And our rule tells us, if we have it in this form, then what we have is that the derivative is going to be equal to the derivative of π of π₯ multiplied by π to the power of π of π₯. And this actually comes from an adaptation of the chain rule.

Okay, great! So we can use this to actually find out what dπ’ dπ₯ is going to be. Well first of all, it is gonna be negative five. And thatβs because if you differentiate negative five π₯, you get negative five. And then this is gonna be multiplied by π to the power of negative five π₯. So great we now have dπ’ dπ₯ and dπ£ dπ₯. So now what we can do is actually go back to our product rule to actually find the derivative of our function.

So first of all, weβre gonna have π to the power of negative five π₯ because thatβs our π’. And then this is gonna be multiplied by two π₯. And thatβs because this is our ππ£ ππ₯. And then this is gonna be plus π₯ squared, which is our π£, and then multiplied by negative five π to the power of negative five π₯ because this is our ππ’ ππ₯. Okay, great! So now letβs rearrange this. And when we do that, we get two π to the power of negative five π₯ multiplied by π₯ minus five π to the power of negative five π₯ multiplied by π₯ squared.

Okay, so now what we can do is actually simplify this by taking out factors. So when we do that, thereβs actually gonna be two results that we can have. So Iβm gonna give you both of those. So the first one that we can find is if we take out π to the power of negative five π₯ multiplied by π₯ as a factor of each of our terms. So if we do that, then inside the parentheses weβre gonna get two minus five π₯. And this is actually our derivative simplified fully. So therefore, we can actually say that the derivative of π¦ equals π to the power of negative five π₯ multiplied by π₯ squared is π to the power of negative five π₯ multiplied by π₯ multiplied by two minus five π₯.

Okay, as I said, thereβs actually another way that we can actually leave our answer. And we get this if we take out negative π to the power of negative five π₯ multiplied by π₯ as a factor this time instead. And we do that so that we can actually have the π₯ term as the first term in our parentheses this time. So what we get is the derivative is actually equal to negative π to the power of negative five π₯ multiplied by π₯, then multiplied by five π₯ minus two. Okay, great! So weβve actually determined the derivative of our function. And we did that using the product rule and then an adaptation of the chain rule.