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Thread: A question on differenciation of trig function(not homework)

  1. #1 A question on differenciation of trig function(not homework) 
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    I am reading on calculus, and am trying to understand the deep subject. However i am facing some problem on understanding the differentiation of a trig function and why when



    I have understood how you get
    by limiting x to 0 for

    The part i cannot grasp is why 3 has to pulled out and placed in front of the differentiated function. My book says it has 3x is a function in the function that can be differenciated further by x.

    Personally, this sounds like chain rule to me. where by where by z = sin(3x)

    In other words i would like to ask if

    Or to put it in other words is Even though there is no x in the function?


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  3. #2  
    Forum Masters Degree organic god's Avatar
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    d/dx (sin(a)) would equal 0 as sin(a) is a constant.

    you are right that diffentiating nested functions like sin(3x) is like a chain rule.

    if we have y = f(g(x)) then dy/dx = df/dg * dg/dx

    so in the case of sin(3x), g(x) = 3x and f(g) = sin (g)

    therefore dy/dx = cos(g) * 3
    = 3 cos(3x)


    everything is mathematical.
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  4. #3  
    Forum Ph.D. Heinsbergrelatz's Avatar
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    the basic derivatives of the Sine function is;
    sin[f(x)]=cos[f(x)]f'(x)
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  5. #4  
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    thank you very much for your response.

    I am sorry not to have it clearer, y = sin(a) . In this scenario can you differentiate by x? Even though there is no x in the function?

    Perhaps this is a weird question to ask but due to chain rule, i am becoming quite confused as to what is dy/dx ? Can i treat it like a fraction? or must it be treated like a function all the time? Cause from the chain rule, it seems like it can work like a fraction, where you can times to whole equation with d(y) to just get dx. However my book strictly says that you should look at dy/dx from the fundamental way of deriving the derivative. (limiting delta x to 0 , for delta(y)/delta(x) )

    Is it possible to explain briefly on what is exactly and how we should treat dy/dx?
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  6. #5  
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    Heinsbergrelatz, thank you for your response, i am aware of the equation but i am more interested on why we must do that. Cause if you prove d/dx (sin3x) simply by using limiting you only get cos(3x) with no constant in front.

    now i know how to it is done by chain rule, but i am confused as to the definition of dy/dx and how one must get a higher derivative in terms of (something)


    I am sorry if my poor english cannot express this 'confusion' properly.
    I appreciate all the responses, thank you
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  7. #6  
    Ots
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    If you differentiate f(x) = sin(a) where is 'a' number and has nothing to do with x, it's like taking the derivative of a constant: if f(x) = 4 then f'(x) = 0 because the derivative of a constant is zero.

    dy/dx is differential notation, read, "The derivative of y with respect to x," where y is a function of x. When working in differential notation, dy/dx can be manipulated algebraically, e.g., if dy/dx = 2, then dy = 2dx.
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  8. #7  
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    I am sorry, i meant a to be like a changing variable like x but just as a.

    Thank you for your response, so that means dy/dx can treated like a fraction. A special fraction that cannot be separated right? Since you cant define dy as anything right? (or dx as a function)
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  9. #8  
    Ots
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    When 'a' can take on any value then it is a variable just like 'x.' But if we have a function of x we call it f(x). If 'a' is a variable like 'x' and f(x) = a then the value of f(x) always equals a and does not depend on any value for x to do so. We then say that f(x) is equal to a constant (in this case 'a'). We don't know what the constant is until a value for it is given. But the function no longer depends on x. It's derivative is then zero.

    dy/dx isn't so much a fraction as it is a function: a change in y due to a change in x, and as I said before it can be treated algebraically since we are using differential notation.
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  10. #9  
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    thank you very much for your response. I understand now. Thank you very much.
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