Step by step solution to "Consider a function z=xy+x(y2+1).Find first order partial derivatives,..."

nagasenaz

Answered question

2021-02-09

Consider a function

z = x y + x (y^{2} + 1)

.Find first order partial derivatives, total differential, and total derivative with respect to x.

Szeteib

Skilled2021-02-10Added 102 answers

Step 1
The function is

z = x y + x (y^{2} + 1)

find the partial derivatives

\frac{\partial z}{\partial x} = y + (y^{2} + 1)

\frac{\partial z}{\partial x} = x + x (2 y)

The partial derivatives are:

\frac{\partial z}{\partial x} = y + y^{2} + 1

\frac{\partial z}{\partial x} = x + 2 x y

Step 2
total differential is given by

d z = \frac{\partial z}{\partial x} d x + \frac{\partial z}{\partial y} d y

Substitute the values

d z = (y + y^{2} + 1) d x + (x + 2 x y) d y

The total differential is:

d z = (y + y^{2} + 1) d x + (x + 2 x y) d y

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} = (y + y^{2} + 1) + (x + 2 x y) \frac{d y}{d x}

The total derivative with respect to x is:

\frac{d z}{d x} = (y + y^{2} + 1) + (x + 2 x y) \frac{d y}{d x}

Jeffrey Jordon

Expert2022-01-30Added 2605 answers

Answer is given below (on video)

Do you have a similar question?

Recalculate according to your conditions!

Didn't find what you were looking for?