Question

encontre as derivadas parciais de primeira ordem para a função z=y . ln x

Answer 1

$z=y\cdot \mathrm{\ell n\,}x$

Para essas derivadas, nem vamos precisar utilizar a regra do produto na sua forma geral. Observe:

$\bullet\;\;\dfrac{\partial z}{\partial x}(x,\,y)=\dfrac{\partial}{\partial x}(y\cdot \mathrm{\ell n\,}x)\\ \\ \\ \dfrac{\partial z}{\partial x}(x,\,y)=y\cdot \dfrac{\partial}{\partial x}(\mathrm{\ell n\,}x)\\ \\ \\ \dfrac{\partial z}{\partial x}(x,\,y)=y\cdot \dfrac{1}{x}\\ \\ \\ \boxed{\begin{array}{c}\dfrac{\partial z}{\partial x}(x,\,y)=\dfrac{y}{x} \end{array}}$


$\bullet\;\;\dfrac{\partial z}{\partial y}(x,\,y)=\dfrac{\partial}{\partial y}(y\cdot \mathrm{\ell n\,}x)\\ \\ \\ \dfrac{\partial z}{\partial y}(x,\,y)=\mathrm{\ell n}(x)\cdot \dfrac{\partial}{\partial y}(y)\\ \\ \\ \dfrac{\partial z}{\partial y}(x,\,y)=\mathrm{\ell n}(x)\cdot 1\\ \\ \\ \boxed{\begin{array}{c}\dfrac{\partial z}{\partial y}(x,\,y)=\mathrm{\ell n\,}x \end{array}}$