Video: Differentiating Root Functions

Find d𝑦/dπ‘₯ if 𝑦 = (6√π‘₯)/7.

01:46

Video Transcript

Find d𝑦 by dπ‘₯ if 𝑦 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 𝑛th root of π‘₯ can be rewritten as π‘₯ to the power of one over 𝑛. Although we don’t write the small two for a square root, that is our value of 𝑛. So we can rewrite the square root of π‘₯ as π‘₯ to the power of one-half. Our expression for 𝑦 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‘Ž and 𝑛, the derivative with respect to π‘₯ of π‘Žπ‘₯ to the 𝑛th power is equal to π‘Žπ‘›π‘₯ to the power of 𝑛 minus one.

So here we go then. We, first of all, multiplied by the exponent 𝑛; that’s one-half. And then, we’ve reduced the exponent by one, giving d𝑦 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‘₯. We’ve found then that d𝑦 by dπ‘₯ is equal to three over seven root π‘₯.

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