Applied calculas

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1. Consider the function y = xx (for x > 0).
a) Why does the derivative rule for xn not apply?
b) Why does the derivative rule for ax not apply?
c) What are the four magic properties of logarithms?
d) Take the ln of both sides. Use the properties of logarithms and implicit differentiation to determine y. Express your answer as a function of x only.

  • Applied calculas -

    You must mean y = x^x or y = xx

    take ln of both sides
    ln y = ln (x^x)
    ln y = x lnx
    now differentiate:

    y'/y = x(1/x) + (1)lnx = 1 + lnx
    y' = y(1 + lnx) = y + lnx
    or
    y' = x^x + lnx

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