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How to determine first partial derivative of ln function ex:

Determine first partial derivative of following function

$$f(x,y,z)=\ln(x+2y+3z)$$

I've been tried,but i stuck on ln derivative. Thanks for your answer.

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You wanna do that by using the rules or using the definition? –  Babak S. Dec 3 '13 at 13:15
    
Using rules @B.S –  Gusti Bimo Marlawanto Dec 3 '13 at 13:16

4 Answers 4

up vote 2 down vote accepted

The derivative of a function $f=\ln g$ is $f'=g'/g$. Let $f(x,y,z)\equiv\ln g(x,y,z)\equiv\ln(x+2y+3z)$, so that $g(x,y,z)=x+2y+3z$. Then: $$f_x(x,y,z)=\frac{1}{x+2y+3z}.$$ We treat all variables other than $x$ as constants. Similarly, $$f_y(x,y,z)=\frac{2}{x+2y+3z},$$ and $$f_z(x,y,z)=\frac{3}{x+2y+3z}.$$

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This is a complete source. +1 –  Babak S. Dec 3 '13 at 13:39

Hint: $f_z=\frac{1}{x+2y+3z}\cdot3$.

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Partial derivative of $f$ with respect to $x$ is $f_x=\frac{1}{x+2y+3z}\times \dfrac{d}{dx}(x+2y+3z)$ and similarly the others.

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To find the partial derivative $\frac{\delta f}{\delta x_i}$ of a function $f(x_1,x_2,...x_n)$, we differentiate $f$ with respect to the variable $x_i$ assuming that $f$ is a function of one variable only, namely $x_i$ and all the other variables are constants.

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