Chapter 4: Q8P (page 191)

If , $z = x^{2} + 2 y^{2}, x = r c o s θ, y = r s i n θ$ , find the following partial derivatives.
${(\frac{\partial z}{\partial x})}_{r}$ $(\frac{\partial z}{\partial x})$ .

Short Answer

Expert verified

The value of provided equation is, $(\frac{\partial z}{\partial x}) = - 2 x$ .

Step by step solution

Step 1: Given data

The given data are listed below

$z = x^{2} + 2 y^{2} . . . (1) x = r \cos θ . . . . (2) y = r \sin θ . . . . . (3)$

Partial differentiation

The procedure of calculating the partial derivative of a function is called partial differentiation. This method is used to get the partial derivative of a function with respect to one variable while keeping the other constant.

Calculation

Square the equation 2 and 3, then add.

$x^{2} + y^{2} = r^{2}$ ...(4)

From equation (1),

$z = x^{2} + 2 (x^{2} + y^{2}) - 2 x^{2} z = - x^{2} + 2 r^{2}$

Take the derivative ofwith respect to ,

role="math" localid="1659095459640" ${(\frac{\partial z}{\partial x})}_{r} = \frac{\partial}{\partial x} (- x^{2} + 2 r^{2}) {(\frac{\partial z}{\partial x})}_{r} = - 2 x$

Therefore, ${(\frac{\partial z}{\partial x})}_{r} = - 2 x$ .

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If , $z = x^{2} + 2 y^{2}, x = r c o s θ, y = r s i n θ$ , find the following partial derivatives.
${(\frac{\partial z}{\partial x})}_{r}$ $(\frac{\partial z}{\partial x})$ .

Short Answer

Step by step solution

Step 1: Given data

Partial differentiation

Calculation

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If , z=x2+2y2,x=rcosθ,y=rsinθ , find the following partial derivatives.(∂z∂x)r(∂z∂x).

Short Answer

Step by step solution

Step 1: Given data

Partial differentiation

Calculation

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Most popular questions from this chapter

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If , $z = x^{2} + 2 y^{2}, x = r c o s θ, y = r s i n θ$ , find the following partial derivatives.
${(\frac{\partial z}{\partial x})}_{r}$ $(\frac{\partial z}{\partial x})$ .