### Video Transcript

Determine the derivative of 𝑦

equals 𝑒 to the power of negative five 𝑥 multiplied by 𝑥 squared.

Now to enable us to actually

determine the derivative, what we’re gonna use is the product rule. And we can actually use the product

rule when we have our function in the form 𝑦 equals 𝑢𝑣. And if we take a look at our

function, it’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 𝑢 𝑑𝑣 𝑑𝑥 plus 𝑣 𝑑𝑢 𝑑𝑥.

So what this means is 𝑢 multiplied

by the derivative of 𝑣 plus 𝑣 multiplied by the derivative of 𝑢. Okay, great! So now that we know the product

rule, let’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 𝑢 and 𝑣 are. So 𝑢 is gonna be equal to 𝑒 to

the power of negative five 𝑥, and 𝑣 is gonna be equal to 𝑥 squared.

Next, what we want to do is

actually differentiate our 𝑢 and 𝑣. So I’m gonna start with 𝑣 because

this is more straightforward. So we can say that d𝑣 d𝑥 is going

to be equal to the derivative of 𝑥 squared. Well this is gonna be equal to two

𝑥, just remind us how we did that. So our exponent multiplied by our

coefficient, so two multiplied by one, and then it’s 𝑥 to the power of, and then we

reduce our exponent by one, so two minus one, which just be one. So we get two 𝑥.

So now we can move on to 𝑢. So if we wanna find d𝑢 d𝑥, well

we’re actually gonna have to use is a general rule to help us here as well. And that’s because 𝑢 is in the

form 𝑦 is equal to 𝑒 to the power of 𝑓 of 𝑥. 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 𝑓 of 𝑥 multiplied by 𝑒 to the power of 𝑓 of 𝑥. And this actually comes from an

adaptation of the chain rule.

Okay, great! So we can use this to actually find

out what d𝑢 d𝑥 is going to be. Well first of all, it is gonna be

negative five. And that’s because if you

differentiate negative five 𝑥, you get negative five. And then this is gonna be

multiplied by 𝑒 to the power of negative five 𝑥. So great we now have d𝑢 d𝑥 and

d𝑣 d𝑥. 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, we’re gonna have

𝑒 to the power of negative five 𝑥 because that’s our 𝑢. And then this is gonna be

multiplied by two 𝑥. And that’s because this is our 𝑑𝑣

𝑑𝑥. And then this is gonna be plus 𝑥

squared, which is our 𝑣, and then multiplied by negative five 𝑒 to the power of

negative five 𝑥 because this is our 𝑑𝑢 𝑑𝑥. Okay, great! So now let’s rearrange this. And when we do that, we get two 𝑒

to the power of negative five 𝑥 multiplied by 𝑥 minus five 𝑒 to the power of

negative five 𝑥 multiplied by 𝑥 squared.

Okay, so now what we can do is

actually simplify this by taking out factors. So when we do that, there’s

actually gonna be two results that we can have. So I’m gonna give you both of

those. So the first one that we can find

is if we take out 𝑒 to the power of negative five 𝑥 multiplied by 𝑥 as a factor

of each of our terms. So if we do that, then inside the

parentheses we’re gonna get two minus five 𝑥. And this is actually our derivative

simplified fully. So therefore, we can actually say

that the derivative of 𝑦 equals 𝑒 to the power of negative five 𝑥 multiplied by

𝑥 squared is 𝑒 to the power of negative five 𝑥 multiplied by 𝑥 multiplied by two

minus five 𝑥.

Okay, as I said, there’s actually

another way that we can actually leave our answer. And we get this if we take out

negative 𝑒 to the power of negative five 𝑥 multiplied by 𝑥 as a factor this time

instead. And we do that so that we can

actually have the 𝑥 term as the first term in our parentheses this time. So what we get is the derivative is

actually equal to negative 𝑒 to the power of negative five 𝑥 multiplied by 𝑥,

then multiplied by five 𝑥 minus two. Okay, great! So we’ve actually determined the

derivative of our function. And we did that using the product

rule and then an adaptation of the chain rule.