Question

If u=x^y then the first order partial derivative of u with respect to y is

Answer 1

$\underline{\textbf{Given:}}$

$\mathsf{u=x^y}$

$\underline{\textbf{To find:}}$

$\textsf{First order partial derivative of u with}$

$\textsf{respect to y}$

$\underline{\textbf{Solution:}}$

$\mathsf{Consider,}$

$\mathsf{u=x^y}$

$\textsf{Take logarithm on bothsides, we get}$

$\mathsf{log\,u=log\,x^y}$

$\mathsf{log\,u=y\,log\,x}$ $\;\;\;(\because\textsf{Using power rule})$

$\textsf{Differentiate partially with respect to 'y'}$

$\mathsf{\dfrac{1}{u}\,\dfrac{{\partial}u}{{\partial}y}=log\,x}$

$\mathsf{\dfrac{{\partial}u}{{\partial}y}=u\;log\,x}$

$\implies\boxed{\mathsf{\dfrac{{\partial}u}{{\partial}y}=x^y\;log\,x}}$