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Math Help - [SOLVED] Difficulty with derivatives

  1. #1
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    [SOLVED] Difficulty with derivatives

    I've got two derivatives, one is logarithmic.

    Okay, here is the first one:

    f(x) = ln (x^2 * e^2x) / (x + ln x)

    I tried doing this and the problem exploded into this huge mess. I know there is a simpler way and in fact this problem is simpler than most problems given I just forget how to simplify this. The best I could do was

    Numerator can = 1 / (x^2 * e^2x)
    Denominator can = x + 1/x

    From this however, the entire thing explodes using the appropriate rules. Isn't there some way to simplify this further?


    Okay phew, part II. Using logarithmic differentiation

    f(x) = (cos(x))^x

    lny = ln(cos(x))^x

    1/y = 1/ x cos(x)

    1/y = x * -sin(x) / cos(x)

    I kinda block here. I must multiply by y on both sides but then the left side is just a "1".

    I get y/y = y * -sin(x) * x / cos(x)

    The y on the right can then be substituted by the original y but what happens on the left side?
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  2. #2
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    e^(i*pi)'s Avatar
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    In part 2 you need to use the chain rule.

    When you differentiated \ln(y) you forgot to multiply by \frac{dy}{dx} as per the chain rule.


    Your expression at the end is actually equal to \frac{dy}{dx}
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