What is the Derivative of cos(sin(x)) and Why is it Different from -sin(sin(x))?

In summary, A new member joins the forum and thanks a user named jamesrc for his help with differentiating cos of sinx using the chain rule. The new member is confused about why df(g)/dg is equal to cos(cosx) instead of -sin(sinx). Another user named Al confirms that the new member is correct and clarifies that jamesrc was differentiating sin of cos rather than cos of sin. The new member apologizes and thanks Al for clearing up the confusion.
  • #1
monet A
67
0
Hi I'm new here, great forum!
On another thread that I can't find I found a message from jamesrc about differentiating cos of sinx that says:

df(g(x))/dx = df(g)/dg * dg(x)/dx by the chain rule, this part I get and I thank james for the help the thing that confuses me is why df(g)/dg = cos(cosx).
Aren't we differentiating df(g) wrt (g) which would give -sin(sinx). What am I missing?
 
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  • #2
You're not missing anything; you are correct: [itex]d/dx [\cos (\sin x)] = - \sin (\sin x) \cos x [/itex]
 
  • #3
Oh thanks Al! :smile:
I just found the thread I read and realized that james was differentiating sin of cos not cos of sin, lol.

:blushing:
 
  • #4
I'm glad you cleared jamesrc's good name! :smile:

And welcome to PF, by the way.
 

Related to What is the Derivative of cos(sin(x)) and Why is it Different from -sin(sin(x))?

What is the derivative of cos(sin x)?

The derivative of cos(sin x) is -sin(sin x) * cos x.

How do you find the derivative of cos(sin x)?

To find the derivative of cos(sin x), you can use the chain rule. First, take the derivative of the outer function, cos x, which is -sin x. Then, multiply it by the derivative of the inner function, sin x, which is cos x. This gives you -sin(sin x) * cos x.

What is the graph of the derivative of cos(sin x)?

The graph of the derivative of cos(sin x) is a sinusoidal wave with a period of 2π and an amplitude of 1. It oscillates between -1 and 1, with a maximum at x=π/2 and a minimum at x=3π/2.

What is the relationship between the derivative of cos(sin x) and the derivative of sin(cos x)?

The derivative of cos(sin x) is equal to the negative of the derivative of sin(cos x). This is because the derivative of cos x is -sin x, and the derivative of sin x is cos x, so the chain rule results in a negative sign in front of the derivative of sin x.

How is the derivative of cos(sin x) used in real life?

The derivative of cos(sin x) is used in various fields such as physics, engineering, and economics. For example, it can be used to analyze the motion of a pendulum or to model the fluctuations of stock prices. It is also used in signal processing to analyze periodic signals and in image processing to enhance image contrast.

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