### 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 πth root of π₯ can be rewritten as π₯ to the power of one over π. Although we donβt 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 ππ₯ to the πth power is equal to πππ₯ to the
power of π minus one.

So here we go then. We, first of all, multiplied by the
exponent π; thatβs one-half. And then, weβve 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₯. Weβve found then that dπ¦ by dπ₯ is
equal to three over seven root π₯.