Find the first derivative of 𝑦 equals nine 𝑥 to the power of six minus seven root 𝑥 with respect to 𝑥.
Well, the first thing we’re gonna do is rewrite our function in the exponent form. And to do that, we’re gonna use one of our exponent rules, which is that if we have 𝑥 to the power of a half, this is equal to root 𝑥. So we’re gonna get 𝑦 is equal to nine 𝑥 to the power of six minus seven 𝑥 to the power of a half. We’ve done this because it can be easier now when we differentiate.
So now, we’re gonna differentiate the function to find our first derivative. So we can say that d𝑦 d𝑥 is going to be equal to. It’s worth saying, at this point, we could either write d𝑦 d𝑥 or we could’ve written 𝑦 prime just to denote the first derivative. So when we differentiate our function, we’re gonna get the first term being 54𝑥 to the power of five.
And just to remind us how we did that, so how we differentiate, what we do is we multiplied the exponent by the coefficient, so six multiplied by nine which gives us our 54. And then, we reduced the exponent by one, so six minus one, which gives us five.
And in the same way, we’ll find the second term. So the second term is going to be negative seven over two 𝑥 to the power of negative a half. And this is because, again, we multiplied the exponent by the coefficient. So we had a half multiplied by seven, which gives us seven over two or seven-halves. And then, we reduce the exponent by one. So we take one away from a half, which gives us negative a half.
So now what we’re gonna do is rewrite in the original form. And we’re gonna use one other exponent rule to help us deal with this as well. And that’s if we have 𝑥 to the power of negative 𝑎, then it’s gonna be equal to one over 𝑥 to the power of 𝑎. So therefore, using both of our laws, we can rewrite our first derivative of 𝑦 equals nine 𝑥 to the power of six minus seven root 𝑥 as 54𝑥 to the power of five minus seven over two root 𝑥.
Now, we got the final term because we had seven over two 𝑥 to the power of negative a half. Well, 𝑥 to the power of a half is just root 𝑥, which we have. And because it was negative a half, we make it on the denominator. So that’s how we got our final answer.