differentiation question

Change the root y into a power form. Which gives you y^1/2.
Bring it to the top and youll get:
x= (y+7)y^-1/2 (Notice the sign change on the power)
Then you can expand the bracket. And hence differentiate.

wouldn't root Y be -1/2 as its the denominator?

wouldn't root Y be -1/2 as its the denominator?

Only when you bring it up into the numerator.

Whilst it's in the denominator you're just rewriting $\sqrt{y}$ as $y^{1/2}$

if you were going to differentiate 1 over root y you would need to rearrange it right? would that be y^-1/2 or y^1/2

You can rearrange it, yes.

You have $\dfrac{y+7}{\sqrt{y}}$

Which is the same as

$\dfrac{y+7}{y^{1/2}}$


And can be rewritten as:

$(y+7)y^{-1/2}$


As @2022 g said in their first post.

is it possible to differentiate it further?

is it possible to differentiate it further?

So far we have rearranged the function to make it easier to differentiate. So we haven’t differentiated yet.
(y+7)y^-1/2 you should expand this equation.
And then you begin to differentiate.

Do you mean "differentiate" or "simplify"? You haven't done any differentiation yet

It is certainly possible to simplify by multiplying out the bracket using the standard rules of indices.

To expand it would it be

y2^-1/2 + 7y^-1/2

You got the “7y^-1/2” right.
Did you make a typo with the “y2^-1/2” because that part is incorrect.

y x y =y^2?

That is correct but we are not multiplying y x y. You’re expanding (y+7)y^-1/2 which gives you:
y x y^-1/2 = y^1/2 and 7 x y^-1/2 = 7y-1/2. ===> y^1/2 +7y^-1/2
When multiplying y variables (as shown for “y x y^-1/2 = y^1/2”).
You would add the powers. 1 + (-1/2) = 1/2

So now you have x= y^1/2 +7y^-1/2
So now you would differentiate.

is it possible to differentiate it further?

y x y^-1/2 = y^1/2 and 7 x y^-1/2 = 7y-1/2. ===> y^1/2 +7y^-1/2
When multiplying y variables (as shown for “y x y^-1/2 = y^1/2”).
You would add the powers. 1 + (-1/2) = 1/2


this just doesn't make sense to me, maths isn't my strongest sorry

Are you happy with multiplying out brackets generally - in particular the rule that a(b + c) = ab + ac ?

You're trying to work out $(y + 7)y^{-1/2}$ which is the same as $y^{-1/2}(y + 7)$. So in this case we have $a = y^{-1/2}, b = y, c = 7$.

Then $ab = y^{-1/2} \times y = y^{1/2}$ and $ac = y^{-1/2} \times 7 = 7y^{-1/2}$ so the final answer is $y^{1/2} + 7y^{-1/2}$.

Btw you keep using the word "differentiate" - is this a typo? Differentiation is an advanced maths technique and has nothing to do with what we're doing here, so I think some of the early attempts to help were a bit confused