Chapter 21: Problem 2

In each case, given \(z=f(x, y)\) find \(z_{x}\) and \(z_{y}\). (a) \(z=x y\) (b) \(z=3 x y\) (c) \(z=-9 y x\) (d) \(z=x^{2} y\) (e) \(z=9 x^{2} y\) (f) \(z=8 x y^{2}\)

Short Answer

Expert verified

Question: Find the partial derivatives of the following functions: a) \(z = xy\) b) \(z = 3xy\) c) \(z = -9yx\) d) \(z = x^2y\) e) \(z = 9x^2y\) f) \(z = 8xy^2\) Answer: a) \(z_x = y\), \(z_y = x\) b) \(z_x = 3y\), \(z_y = 3x\) c) \(z_x = -9y\), \(z_y = -9x\) d) \(z_x = 2xy\), \(z_y = x^2\) e) \(z_x = 18xy\), \(z_y = 9x^2\) f) \(z_x = 8y^2\), \(z_y = 16xy\)

Step by step solution

Identify the function

Here, the function is given as \(z = xy\).

Find the partial derivatives

To find the partial derivatives \(z_x\) and \(z_y\), take the derivative of the function with respect to x and y, keeping the other variable constant. \(z_x = \frac{\partial z}{\partial x} = \frac{\partial (xy)}{\partial x} = y\) \(z_y = \frac{\partial z}{\partial y} = \frac{\partial (xy)}{\partial y} = x\) #b)

Identify the function

Here, the function is given as \(z = 3xy\).

Find the partial derivatives

To find the partial derivatives \(z_x\) and \(z_y\), take the derivative of the function with respect to x and y, keeping the other variable constant. \(z_x = \frac{\partial z}{\partial x} = \frac{\partial (3xy)}{\partial x} = 3y\) \(z_y = \frac{\partial z}{\partial y} = \frac{\partial (3xy)}{\partial y} = 3x\) #c)

Identify the function

Here, the function is given as \(z = -9yx\).

Find the partial derivatives

To find the partial derivatives \(z_x\) and \(z_y\), take the derivative of the function with respect to x and y, keeping the other variable constant. \(z_x = \frac{\partial z}{\partial x} = \frac{\partial (-9yx)}{\partial x} = -9y\) \(z_y = \frac{\partial z}{\partial y} = \frac{\partial (-9yx)}{\partial y} = -9x\) #d)

Identify the function

Here, the function is given as \(z = x^2y\).

Find the partial derivatives

To find the partial derivatives \(z_x\) and \(z_y\), take the derivative of the function with respect to x and y, keeping the other variable constant. \(z_x = \frac{\partial z}{\partial x} = \frac{\partial (x^2y)}{\partial x} = 2xy\) \(z_y = \frac{\partial z}{\partial y} = \frac{\partial (x^2y)}{\partial y} = x^2\) #e)

Identify the function

Here, the function is given as \(z = 9x^2y\).

Find the partial derivatives

To find the partial derivatives \(z_x\) and \(z_y\), take the derivative of the function with respect to x and y, keeping the other variable constant. \(z_x = \frac{\partial z}{\partial x} = \frac{\partial (9x^2y)}{\partial x} = 18xy\) \(z_y = \frac{\partial z}{\partial y} = \frac{\partial (9x^2y)}{\partial y} = 9x^2\) #f)

Identify the function

Here, the function is given as \(z = 8xy^2\).

Find the partial derivatives

To find the partial derivatives \(z_x\) and \(z_y\), take the derivative of the function with respect to x and y, keeping the other variable constant. \(z_x = \frac{\partial z}{\partial x} = \frac{\partial (8xy^2)}{\partial x} = 8y^2\) \(z_y = \frac{\partial z}{\partial y} = \frac{\partial (8xy^2)}{\partial y} = 16xy\)

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In each case, given \(z=f(x, y)\) find \(z_{x}\) and \(z_{y}\). (a) \(z=x y\) (b) \(z=3 x y\) (c) \(z=-9 y x\) (d) \(z=x^{2} y\) (e) \(z=9 x^{2} y\) (f) \(z=8 x y^{2}\)

Short Answer

Step by step solution

Identify the function

Find the partial derivatives

Identify the function

Find the partial derivatives

Identify the function

Find the partial derivatives

Identify the function

Find the partial derivatives

Identify the function

Find the partial derivatives

Identify the function

Find the partial derivatives

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