nagasenaz

2021-02-09

Consider a function $z=xy+x\left({y}^{2}+1\right)$.Find first order partial derivatives, total differential, and total derivative with respect to x.

Szeteib

Step 1
The function is $z=xy+x\left({y}^{2}+1\right)$
find the partial derivatives
$\frac{\partial z}{\partial x}=y+\left({y}^{2}+1\right)$
$\frac{\partial z}{\partial x}=x+x\left(2y\right)$
The partial derivatives are:
$\frac{\partial z}{\partial x}=y+{y}^{2}+1$
$\frac{\partial z}{\partial x}=x+2xy$
Step 2
total differential is given by $dz=\frac{\partial z}{\partial x}dx+\frac{\partial z}{\partial y}dy$
Substitute the values
$dz=\left(y+{y}^{2}+1\right)dx+\left(x+2xy\right)dy$
The total differential is:
$dz=\left(y+{y}^{2}+1\right)dx+\left(x+2xy\right)dy$
Step 3
The total derivative with respect to x is given by
$\frac{\partial z}{\partial x}=\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}\frac{\partial y}{\partial x}$
Substitute the values
$\frac{\partial z}{\partial x}=\left(y+{y}^{2}+1\right)+\left(x+2xy\right)\frac{dy}{dx}$
The total derivative with respect to x is:
$\frac{dz}{dx}=\left(y+{y}^{2}+1\right)+\left(x+2xy\right)\frac{dy}{dx}$

Jeffrey Jordon