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Math Help - Simple Differentiation

  1. #1
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    Exclamation Simple Differentiation

    How do I exactly differentiate 3^(5x)

    Thanxx a lot!
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  2. #2
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    Quote Originally Posted by xwrathbringerx View Post
    How do I exactly differentiate 3^(5x)

    Thanxx a lot!
    a^x=(x')a^x \ln a
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  3. #3
    MHF Contributor Calculus26's Avatar
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    The general fromula for f= a^(u(x))

    f ' = ln(a) a^(u(x))du/dx

    For 3^(5x)

    f ' = ln(3)3^(5x)5 = 5ln(3)3^(5x)
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  4. #4
    MHF Contributor Calculus26's Avatar
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    doesn't make sense as x ' = 1

    (a^x ) ' = ln(a)*a^x period

    for a^(u(x) use the chain rule to obtain

    f ' = ln(a) a^(u(x))du/dx
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  5. #5
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    A general method for problems like this is to take the logarithm of both sides. If y= 3^{5x} then ln(y)= ln(3^{5x})= 5x ln(3). Now differentiate both sides with respect to x.

    The derivative of ln(y) with respect to y is \frac{1}{y} but since y itself is a function of x, by the chain rule, the derivative of ln(y) with respect to x is \frac{1}{y}\frac{dy}{dx}.

    \frac{1}{y}\frac{dy}{dx}= 5 ln(3) so \frac{dy}{dx}= 5 ln(3) y= 5 ln(3) 3^{5y}.
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