Implicit differentiation/ partial differentiation

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July 26th 2010, 04:35 PM
CookieC

Implicit differentiation/ partial differentiation

How do i find the partial derivative in terms of Y from the equation z = x^y even if X is held as a constant i dont know how to differentiate it :S
July 26th 2010, 04:36 PM
CookieC

Actually, is the answer just zero?
July 26th 2010, 05:24 PM
bondesan

Follow these steps, where $z_y$ means the derivative of z with respect of y.

$\displaystyle{z=x^y}$

$\displaystyle{ln(z) = y\cdot ln(x)}$

$\displaystyle{\frac{\partial}{\partial y} ln (z) = \frac{\partial}{\partial y}\left(y\cdot ln (x)\right)}$

$\displaystyle{\frac{z_y}{z} = ln (x)}$

$\displaystyle{z_y = z\cdot ln (x) = x^y\cdot ln (x)}$

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