Question Video: Differentiating a Combination of Root Functions Mathematics • Second Year of Secondary School

Video Transcript

Find the derivative of negative two times the square root of 𝑥 minus seven 𝑥 with respect to 𝑥.

We’re asked to find the derivative of an expression with respect to 𝑥. And we know how to differentiate negative seven 𝑥 with respect to 𝑥. However, we don’t know how to differentiate negative two root 𝑥 directly. So to differentiate this expression, we’re going to want to rewrite root 𝑥 in a form which we can differentiate. Luckily, we can do this by using our laws of exponents.

We recall the square root of 𝑥 is the same as 𝑥 to the power of one-half. So by using this, we can rewrite this as the derivative of negative two 𝑥 to the power of one-half minus seven 𝑥 with respect to 𝑥. And now we can see we can differentiate this term by term by using the power rule for differentiation. We recall this tells us for any real constants 𝑎 and 𝑛, the derivative of 𝑎𝑥 to the 𝑛th power with respect to 𝑥 is equal to 𝑎 times 𝑛 times 𝑥 to the power of 𝑛 minus one. We multiply by our exponent of 𝑥 and then reduce this exponent by one. And we want to apply this to each term separately.

In our first term, we want our value of 𝑎 equal to negative two and our value of 𝑛 equal to one-half. So by using the power rule for differentiation, we multiply by our exponent one-half and then reduce this exponent by one. This gives us negative two times one-half multiplied by 𝑥 to the power of one-half minus one.

We then want to apply this to our second term. And it might be easier to think of negative seven 𝑥 as negative seven 𝑥 to the first power. Then we can see that our value of 𝑎 is negative seven and the value of our exponent of 𝑥 is one. So applying the power rule for differentiation on our second term, we want to multiply by our exponent of 𝑥, which is one, and then reduce this exponent by one. This gives us negative seven times one multiplied by 𝑥 to the power of one minus one.

Now, we can start simplifying this expression. First, the coefficient in our first term, negative two times one-half, is equal to negative one. And our exponent one-half minus one is equal to negative one-half. So our first term is negative 𝑥 to the power of negative one-half. Then, in our second term, the coefficient negative seven times one is equal to negative seven. And then in our exponent of 𝑥, we have one minus one, which is equal to zero. And of course, instead of adding negative seven times 𝑥 to the zeroth power, we can instead just subtract seven 𝑥 to the zeroth power. But then 𝑥 to the zeroth power is just equal to one. So we can just remove this altogether.

We could leave our answer like this. However, we can rewrite 𝑥 to the negative one-half by using our laws of exponents. We need to recall that raising a number to the negative exponent is the same as dividing by that same number raised to the positive exponent. So 𝑥 to the power of negative one-half is the same as dividing by 𝑥 to the power of one-half, which is of course equal to one divided by root 𝑥.

So by using our laws of exponents and rearranging our two terms, we got negative seven minus one divided by root 𝑥, which is our final answer. Therefore, by using our laws of exponents and the power rule for differentiation, we were able to show the derivative of negative two root 𝑥 minus seven 𝑥 with respect to 𝑥 is equal to negative seven minus one divided by root 𝑥.