# Question Video: Differentiating Root Functions Mathematics • Higher Education

Find dπ¦/dπ₯ if π¦ = (6βπ₯)/7.

01:44

### Video Transcript

Find dπ¦ by dπ₯ if π¦ equals six root π₯ over seven.

In order to answer this question, we need to think about the alternative ways that we can express a square root. We know from our laws of exponents that the πth root of π₯ can be rewritten as π₯ to the power of one over π. Although we donβt write the small two for a square root, that is our value of π. So we can rewrite the square root of π₯ as π₯ to the power of one-half. Our expression for π¦ can therefore be rewritten as six-sevenths π₯ to the power of one-half. And we see that we have a general power term. We can differentiate this using the power rule of differentiation which tells us that for real constants π and π, the derivative with respect to π₯ of ππ₯ to the πth power is equal to πππ₯ to the power of π minus one.

So here we go then. We, first of all, multiplied by the exponent π; thatβs one-half. And then, weβve reduced the exponent by one, giving dπ¦ by dπ₯ equals six-sevenths multiplied by one-half multiplied by π₯ to the power of one-half minus one. We can simplify the coefficient by cancelling a shared factor of two. And then one-half minus one is equal to negative one-half. We then recall another of our laws of exponents which is that a negative exponent defines a reciprocal. So π₯ to the power of negative one-half is one over π₯ to the power of a half or one over the square root of π₯. Weβve found then that dπ¦ by dπ₯ is equal to three over seven root π₯.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.