### Video Transcript

Find dπ¦ by dπ₯ if π¦ equals two π₯ to the power of negative seven.

In this question, we were asked to find dπ¦ by dπ₯, which we recall means the first derivative of π¦ with respect to π₯. So weβre going to answer this question using differentiation. When we look at our function π¦, we see that it consists of just a single term in π₯. It is a general polynomial term, two multiplied by π₯ to the power of negative seven.

We therefore need to recall the power rule of differentiation, which tells us how to differentiate general polynomial terms. In its most straightforward form, the power rule tells us that the derivative with respect π₯ of π₯ to some power π, where π is a real number, is equal to π multiplied by π₯ to the power of π minus one. We multiply by the original exponent and then reduce the exponent by one.

If we introduce the coefficient π, so weβre looking to find the derivative with respect to π₯ of ππ₯ to the πth power. Then we recall that the rules of differentiation tell us that this is just equal to π multiplied by the derivative with respect π₯ of π₯ to the πth power. So by applying the first rule, this is equal to πππ₯ to the power of π minus one.

In this question then, the value of π, the coefficient, is two and the value of π, the exponent, is negative seven. Letβs apply the power rule of differentiation then. We have that dπ¦ by dπ₯ will be equal to two. Then we multiply by the original exponent, which was negative seven. And then we reduce the exponent by one, giving π₯ to the power of negative seven minus one.

Simplifying the coefficient, two multiplied by negative seven is negative 14. And then simplifying the exponent, negative seven minus one is negative eight. So we have that dπ¦ by dπ₯ is equal to negative 14 multiplied by π₯ to the power of negative eight.

Be careful here because weβre working with a negative exponent. And remember, we need to reduce the exponent by one. A common mistake would be to make the new exponent negative six rather than negative eight.

So by applying the power rule of differentiation, weβve found that if π¦ is equal to two π₯ to the power of negative seven, then dπ¦ by dπ₯ is equal to negative 14π₯ to the power of negative eight.