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How to differentiate y=x(4 - x^2)^0.5?

Hi, I am really confused on how you are meant to differentiate this: y=x(4 - x^2)^0.5? Can someone please help. Many thanks

Reply 1

Notnek

Original post by Bertybassett

Hi, I am really confused on how you are meant to differentiate this: y=x(4 - x^2)^0.5? Can someone please help. Many thanks

What have you tried?

Reply 2

Bertybassett

I tried using the chain rule, using u=4-x^2 and got a mess. I just don't really know where to start or what method to use.

Original post by Notnek

What have you tried?

Reply 3

RDKGames

Study Forum Helper

Original post by Bertybassett

Hi, I am really confused on how you are meant to differentiate this: y=x(4 - x^2)^0.5? Can someone please help. Many thanks

Combination of product rule and chain rule.

Alternatively, you can make your life easy by squaring both sides to get

y^2 = x^2(4-x^2)

then differentiate both sides (making use of implicit differentiation on the LHS) and hence rearrange for dy/dx.

Reply 4

Notnek

Original post by Bertybassett

I tried using the chain rule, using u=4-x^2 and got a mess. I just don't really know where to start or what method to use.

This is a product of functions so you need to use the product rule with x and (4 - x^2)^0.5 and for the latter you need to use the chain rule.

Please have a go at this and post all your working if you get stuck.

Reply 5

Bertybassett

what do you mean by "combination of product and chain rule"? How do you do this?

Original post by RDKGames

Combination of product rule and chain rule.

Alternatively, you can make your life easy by squaring both sides to get

y^2 = x^2(4-x^2)

then differentiate both sides (making use of implicit differentiation on the LHS) and hence rearrange for dy/dx.

Reply 6

RDKGames

Study Forum Helper

Original post by Bertybassett

what do you mean by "combination of product and chain rule"? How do you do this?

It's of the form

y=uv

. So

y' = u'v + uv'

.

We can let

u=x

and

v=\sqrt{4-x^2}

. Clearly,

x'=1

. What is

v'

? Use the chain rule for that.

Then sub them all into the chain rule eqn on the second sentence.

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How to differentiate y=x(4 - x^2)^0.5?

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