# Derivative of $\sqrt{4 \sin x + 2}$ at $0$

If $f(x) = \sqrt{ 4\sin x + 2 }$, then $f'(0) =$?

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Is this homework? What part of this are you having trouble with? – Brandon Carter Dec 22 '10 at 1:28
well this isnt homework for me its for my friends project he did all of it except that one cuz he doesn't know what to do and i haven't taken calculus yet to help him – Ronnie Dec 22 '10 at 1:35

Do you know the chain rule? What have you tried?

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By definition,

$$f'(0)=\lim_{h\to0}\frac{f(h)-f(0)}{h}=\lim_{h\to0}\frac{\sqrt{4\sin(h)+2}-\sqrt2}{h}.$$

You can "rationalize" the numerator, use the fact that a limit of a product is the product of the limits, use continuity of the square root and sine functions, and use the fact that $\lim_{h\to0}\frac{\sin(h)}{h}=1$ to finish. Of course, you would get the same answer by deriving a general formula for $f'(x)$ using the chain rule and plugging in $x=0$, as Ross hints.

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my friend asks for the derivative of or sqrt(4sinx+2) any helps/tips for him? – Ronnie Dec 22 '10 at 1:50
@Ronnie: Unless you can tell us everything your friend knows, it would be easier for him to be asking. I am a bit perplexed by your question, as you have already received some tips, and you have not responded to them. Ross mentioned that the chain rule is relevant; if your friend does not see why, then some elaboration on the question would be helpful. Does your friend know how to take the derivative of the square root and sine functions? My answer outlines a method of directly computing $f'(0)$, so again, I'm confused by your question as to whether I can give any tips. – Jonas Meyer Dec 22 '10 at 1:57
im sorry my friend is confused too he needs fto find the derivitive of sqrt(4sinx+2) and i cant hhelp him because i dont take calculus yet and he is unfamiliar with the equation you gave above – Ronnie Dec 22 '10 at 2:03
@Ronnie: That equation is just the definition of the derivative. If he does not know the definition of the derivative, then oddly enough he could still do this problem if he knows the chain rule and the formulas for the derivatives of sine and square root functions. You have still not answered Ross's or my questions. Does your friend know the chain rule and the formulas for the derivatives of the sine and square root functions? What has your friend tried? Why is your friend not asking the question? – Jonas Meyer Dec 22 '10 at 2:08

$$f'(x)=\frac{4\cos x}{2\sqrt{4\sin x+2}}$$

$$f'(0)=\frac{4\cdot1}{2\sqrt{4\cdot0+2}}=\frac{4}{2\sqrt{2}}=\sqrt{2}$$

what is the difficulty? It does not even involve a $\frac{0}{0}$ like $\sin{x}/x$ does.

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