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Math Help - Proving the derivative.

  1. #1
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    Proving the derivative.

    "Use the limit definition of the derviative to prove the derivative for the function y = tan x is sec^2 x

    (You'll need to use the lim as h goes to 0 (sin h/h) as well as the identity for the tangent of the sum of two angles)"

    I know that that's the derivative, but I have no idea how to prove it. Please help!
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  2. #2
    Senior Member apcalculus's Avatar
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    Write the tangent as ratio of sine over cosine, and use the definition:

    f'(a)=\lim_{h \to 0} \frac{f(a+h)-f(a)}{h}

    Note that:
    f(a+h) = \tan(a+h) = \frac{\sin(a+h)}{\cos(a+h)}

    and also:
    f(a)=\tan(a) = \frac{\sin(a)}{\cos(a)}
    The numerator becomes:
    f(a+h) - f(a) = \frac{\sin(a+h)}{\cos(a+h)} - \frac{\sin(a)}{\cos(a)}

    Put these in the common denominator to get:
    \frac{\cos(a) \sin(a+h) - \cos(a+h) \sin(a)}{\cos(a) \cos(a+h)}

    I hope this is helpful.
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  3. #3
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    Does that cancel to equal sec^2 x? I can't figure out how it cancels if that is the case.
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