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Math Help - Differentiation

  1. #1
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    Differentiation

    Hi

    I would appreciate if someone could check my work.

    Differentiate the following functions

    a)
    f(x) = ln (tan 3x + cos 5x)^2

    f'(x) = (1/(tan 3x + cos 5x)^2) (2(sec^2^x + -sin 5x))

    f'(x) = (2sec^2 3x - 2sin 5x)/((tan 3x + cos 5x)^2)


    b)
    f(x) = e^x^3 + 2ln x) cos (1/x^2))

    f'(x) = (e^x^3 + 2 ln x) ((-sin (1/x^2))(-2x)) + (e^x^3 + 2/x) (cos (1/x^2))


    c)
    f(x) = sin ^ tan x

    ln f(x) = ln (sin ^ tanx)
    ln f(x) = tanx (ln sin)
    f'(x) (1/f(x)) = ((sec^2 x) (ln sin) + (1/sin)(tanx))
    f'(x) = sin ^tan x ((sec^2x)(ln sin) + (1/sin)(tan x))

    Thanks in advance
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  2. #2
    MHF Contributor

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    Quote Originally Posted by calcbeg View Post
    Hi

    I would appreciate if someone could check my work.

    Differentiate the following functions

    a)
    f(x) = ln (tan 3x + cos 5x)^2

    f'(x) = (1/(tan 3x + cos 5x)^2) (2(sec^2^x + -sin 5x))
    That should be 2(3sec^3(3x)- 5sin(5x))

    f'(x) = (2sec^2 3x - 2sin 5x)/((tan 3x + cos 5x)^2)


    b)
    f(x) = e^x^3 + 2ln x) cos (1/x^2))

    f'(x) = (e^x^3 + 2 ln x) ((-sin (1/x^2))(-2x)) + (e^x^3 + 2/x) (cos (1/x^2))
    What happened to the derivative of e^(x^3)? rather that the last e^x^3, you shouled have (e^(x^3))(2x^2).


    c)
    f(x) = sin ^ tan x

    ln f(x) = ln (sin ^ tanx)
    ln f(x) = tanx (ln sin)
    f'(x) (1/f(x)) = ((sec^2 x) (ln sin) + (1/sin)(tanx))
    f'(x) = sin ^tan x ((sec^2x)(ln sin) + (1/sin)(tan x))

    Thanks in advance
    "sin^tanx" doesn't make sense. Do you mean sin^tan(x) (x)?
    Don't forget to differentiate the sin x also.
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