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What is the derivative of this function?

$$ f(x)= \cos \bigg(\frac{\sin x}{x}\bigg) $$

Jeel Shah
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Aysel
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3 Answers3

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I’m assuming that you mean that $$f(x)=\cos\left(\frac{\sin x}x\right)\;.$$ Just apply the chain rule. At the outermost level this is a cosine, so differentiate it as such:

$$f\,'(x)=-\sin\left(\frac{\sin x}x\right)\cdot\left(\frac{\sin x}x\right)'\;,$$

where the prime denotes the derivative with respect to $x$. Now you need $\left(\frac{\sin x}x\right)'$; the function is a quotient, so use the quotient rule:

$$\left(\frac{\sin x}x\right)'=\frac{x(\sin x)'-(\sin x)x'}{x^2}=\frac{x\cos x-\sin x}{x^2}\;.$$

Now just put the pieces together.

Brian M. Scott
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You work from the outside in, slowly apply the chain rule. First, what is the derivative of $\cos$? Well that's $-\sin x$. Thus far we have

$$f'(x)=-sin\bigg(\frac{\sin x}{x}\bigg) \bigg(\frac{\sin x}{x}\bigg)'$$

where $\bigg(\frac{\sin x}{x} \bigg)'$ is the derivative of the 'stuff' on the inside (this came from the chain rule). So we need the derivative of $\frac{\sin x}{x}$. Use the quotient rule. You will get $$\frac{x \cos x-\sin x}{x^2}$$ Now we multiply these to get the answer $$f'(x)=-\sin\bigg(\frac{\sin x}{x} \bigg)\frac{x \cos x-\sin x}{x^2}$$

mathematics2x2life
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HINT Use chain rule to differentiate. Remember the general "formula" for the chain rule is: $$f' = g'(h(x)) * h'(x)$$

In this case, your $g(x)$ is $\cos x$ and your $h(x)$ is $\frac{\sin x}{x}$. You will also have to use to quotient rule to differentiate your $\frac{\sin x}{x}$ term. Recall that quotient rule is:

$$\frac{f'g - fg'}{g^2}$$

So in effect, you have

$$f' = g'(h(x)) * \frac{f'g - fg'}{g^2} $$

HINT 2 The derivative of $\cos x$ is $-\sin x$ and the derivative of $\sin x$ is $\cos x$

Can you take it from here?

Jeel Shah
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    Yess I understand everything Thanks for clear explanation)) – Aysel Nov 12 '13 at 18:31
  • No problem! If you liked the answer, you can upvote it or you can accept it by clicking the check mark. If you like the answers of the other users then you can upvote them too. – Jeel Shah Nov 12 '13 at 18:36
  • I have not enough reputation so, I cant upvote them:( – Aysel Nov 12 '13 at 18:41
  • Right. Well, you can accept one answer so this question is laid to rest and then you can try to gain some rep so you can upvote the other answers. Don't worry about it too much. – Jeel Shah Nov 12 '13 at 18:54