Logarithmic Differentiation

Calculus Tutorial

Logarithmic Differentiation

Intro

You need to use Logarithmic Differentiation when you want to find the derivative of a function that is in the (x^x) form. You start out by equating y to any function that is (x^x). You then apply logarithmic rules to the function by encasing “y” with ln and bringing the power of “x” down in front of your base “x”. You then encase the base “x” with ln and then you can start deriving. You need to apply the product rule to solve the rest and multiply everything by the original x function to complete the derivative.

Sample Problem

y = x^x find the derivative

Solution

[ln(y) = xln(x)]d/dx product rule u'(x)v(x)+u(x)v'(x)

1/y * dy/dx = 1*ln(x) + x*(1/x)

(dy/dx)/y = ln(x) + 1

y[(dy/dx)/y = [ln(x) + 1]]

dy/dx = y[ln(x) + 1]

dy/dx = x^x[ln(x) + 1]



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