# Question Video: Using the Product Rule Mathematics • Higher Education

Using the product rule, find (d/dπ₯)(π₯Β²π^(βπ₯)).

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### Video Transcript

Using the product rule, find the derivative of π₯ squared π to the power of negative π₯.

Well, we know that we can use the product rule because actually we have our function in the form π¦ equals π’π£. And what the product rule tells us is that if we have a function in this form, then the derivative is gonna be equal to π’ dπ£ dπ₯ plus π£ dπ’ dπ₯. So what that tells us is that π’, so the first part of our function, multiplied by the derivative of our π£ plus π£ multiplied by the derivative of our π’.

So the first thing we do with our question is actually identify what our π’ and π£ are gonna be. Our π’ is gonna be π₯ squared, our π£ is gonna be equal to π to the power of negative π₯. So therefore, our dπ’ dπ₯ is gonna be equal to two π₯. And we got that because we use the same rule that we do when weβre differentiating any term. And thatβs actually we multiply the coefficient, which was one, by the exponent, which was two, which gives us our two. And then, we reduce the exponent by one. So weβve got two minus one, which is one. So we get two π₯.

Okay, great, so now letβs find dπ£ dπ₯. Well, dπ£ dπ₯ is just gonna be equal to negative π to the power of negative π₯. So what we know about π is that if you actually differentiate π to the power of π₯, you get π to the power of π₯. Thatβs one of its sort of special properties.

But how we actually got our answer is using the chain rule cause we know that if you have π to the power of π of π₯, then thisβs gonna be equal to the derivative of π of π₯ multiplied by π to power of π of π₯. Well, in our case, our π of π₯ was negative π₯, where the derivative of negative π₯ is just negative one. So therefore, itβll be negative one multiplied by π to the power of negative π₯, which is what we got.

So now, what we can do is actually bring everything together using the product rule to actually find the derivative of π₯ squared π to the power of negative π₯. So what weβre first gonna have is π₯ squared multiplied by negative π to the power of negative π₯. And thatβs our π’ and our dπ£ dπ₯, so π’ multiplied by dπ£ dπ₯. And then, this is plus π to the power of negative π₯ multiplied by two π₯ because this is our π£ dπ’ dπ₯.

So then, if we actually tidied this up, we can actually say that the derivative of π₯ squared π to the power of negative π₯ is gonna be equal to two π₯ π to the power of negative π₯ minus π₯ squared π to the power of negative π₯.