Question

Differentiate (sqrtx)cos(sqrtx).

Answer 1

Chain rule and product rule...
\[\Large \frac{d}{dx}\left[\sqrt{x}\cos(\sqrt x)\right]\]

Answer 2

As a refresher:
Chain Rule:
\[\Large \frac{d}{dx}{f\left(g(x)\right)}= f'\left(g(x)\right)g'(x)\]

Product Rule:
\[\Large \frac{d}{dx}\left[f(x)g(x)\right]= f'(x)g(x) + f(x) g'(x)\]

Answer 3

First apply the product rule, though...

Answer 4

Would it be 1/2x^(-1/2)cosx^(1/2)+x^1/2(1/2-sinx^-1/2)

Answer 5

Yikes... that's messy... lol, give me a sec to digest it ^_^

Answer 6

Mind telling me how you got the second term?

Answer 7

x^1/2(1/2-sinx^-1/2)

Answer 8

you leave x^1/2 by itself and isn't the derivative of cos(sqrtx) -sin(1/2^x^(-1/2)

Answer 9

really cannot understand that... could you draw it instead?
There's the draw option... what's the derivative of \(\large \cos \sqrt x\)

Answer 10

I meant draw it, you know, free drawing...so that you can use symbols...

Answer 11

Is this what you want to say:
\[\Large \sin \left(\frac1{2\sqrt x }\right)\]?

Answer 12

Or rather, negative of that, sorry.

Answer 13

\[-\sin \frac{ 1 }{ 2} \chi ^{-1/2}\]

Answer 14

yes that is what I am trying to say

Answer 15

Well then, it's wrong ^_^

Answer 16

Chain rule...

Answer 17

so what is the correct answer

Answer 18

Let's find out together.

Answer 19

\[\Large \frac{d}{dx}\cos(\sqrt x)\]

Answer 20

Again, chain rule
\[\Large \frac{d}{dx}{f\left(g(x)\right)}= f'\left(g(x)\right)g'(x)\]
Take f to be cos(x)
and g to be sqrt(x)

Answer 21

\[-\sin (\sqrt{x})(-\frac{ 1 }{ 2}x ^{-1/2})\]

Answer 22

Still something wrong...

Answer 23

\[-\sin(\sqrt{x})(\frac{ 1 }{ 2}x ^{-1/2})\]

Answer 24

Better.
\[\Large -\frac1{2\sqrt x}\sin(\sqrt x)\]

Answer 25

So, use that derivative when evaluating the derivative of \(\large \cos \sqrt x\)

And what's your final answer?

Answer 26

\[\frac{ 1 }{2 \sqrt{x}}(\cos \sqrt{x})+(-\frac{ 1 }{ 2\sqrt{x} }\sin \sqrt{x})\]

Answer 27

You forgot something in the second term... (and yes, differentiating can be this dizzying)

Answer 28

did I forget to do the chain rule

Answer 29

Well, you forgot a certain factor...

Answer 30

what do you mean?