Find d𝑦 by d𝑥 if 𝑦 equals six root 𝑥 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 𝑥 can be rewritten as 𝑥 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 𝑥 as 𝑥 to the power of one-half. Our expression for 𝑦 can therefore be rewritten as six-sevenths 𝑥 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 𝑎 and 𝑛, the derivative with respect to 𝑥 of 𝑎𝑥 to the 𝑛th power is equal to 𝑎𝑛𝑥 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 d𝑥 equals six-sevenths multiplied by one-half multiplied by 𝑥 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 𝑥 to the power of negative one-half is one over 𝑥 to the power of a half or one over the square root of 𝑥. We’ve found then that d𝑦 by d𝑥 is equal to three over seven root 𝑥.