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I have recently read that

$$\frac{d}{dx}\sin(x)=\cos(x)$$

But,unfortunately I have not been able to make out what it exactly means that the rate of change of $\sin(x)$ is $\cos(x)$?

What does it simply mean that rate of change of $\sin(x)$ is $\cos(x)$?

I am a beginner in this branch of mathematics.

Any help is welcome.

choco_addicted
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Soham
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  • Context would be nice. Have you learned any calculus? Are you comfortable, for example, with the claim that $\frac{d}{dx}x^2 = 2x$, or any claim involving $\frac{d}{dx}$? – pjs36 Feb 26 '16 at 06:22
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    To get a visual idea, you might want to look at the graphs of $\sin x$ and $\cos x$, and compare the slope of the tangents to $\sin x$ graph with the value of $\cos x$, especially at points $x = \frac{n\pi}{2}$. – vnd Feb 26 '16 at 06:23
  • You have it exactly right, the rate of change of sin(x) is indeed cos(x). – Kaynex Feb 26 '16 at 06:29
  • Not directly related to the question, but I wonder if the original poster would also appreciate a link a good set of introductory calculus notes. A more in depth discussion would be more likely to give understanding in this case. – Eric Thoma Feb 26 '16 at 06:50
  • @ClaudeLeibovici: I'm not convinced this question is a duplicate of the cited question. It's plausible to me that pjs36's question is on point: that the OP is seeking to understand how the derivative of $\sin x$ is some other function. (To many students, it seems to answer one question with another.) More context is needed, to be sure; I might well have voted to put the question on hold for that reason. – Brian Tung Feb 26 '16 at 23:39
  • @tatan: In fact, the derivative—which in this case you may rightly view as a rate of change—of $\sin x$ is $\cos x$. That is to say, the rate of change of $\sin x$ at $x = 0$ is $1$, which happens to be $\cos 0$. The rate of change of $\sin x$ at $x = \pi/4$ at $x = 0$ is $\sqrt{2}{2}$, which happens to be $\cos \pi/4$; the rate of change at $x = \pi/3$ is $1/2$, which happens to be $\cos \pi/3$; and in general, the rate of change at $x = x_0$ is whatever $\cos x_0$ happens to be. (Of course, calculus shows that there's nothing "happens to be" about it—that this is the rule.) – Brian Tung Feb 26 '16 at 23:43

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What this means is that the slope of the line tangent to the graph of $\sin(x)$ at a point $(x_0, \sin(x_0))$ is equal to $\cos(x_0)$.

SplitInfinity
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