Chapter 4: Q14P (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.
${(\frac{\partial z}{\partial θ})}_{y}$ .

Short Answer

Expert verified

The value of provide equation is ${(\frac{\partial z}{\partial θ})}_{y} = - 2 r^{2} \sin 2 θ$ .

Step by step solution

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

From equation 1 and 2,

$\begin{array}{l} z = (r \cos θ)^{2} + 2 y^{2} \\ z = r^{2} \cos^{2} θ + 2 y^{2} \end{array}$

Take the partial derivative of 'z' with respect to ' $θ$ ' in above equation;

role="math" localid="1659092547223" ${(\frac{\partial z}{\partial θ})}_{y} = \frac{\partial}{\partial θ} [r^{2} \cos^{2} θ + 2 y^{2}] = 2 r^{2} (\sin θ \cdot \cos θ) {(\frac{\partial z}{\partial θ})}_{y} = 2 r^{2} \sin 2 θ$

Hence,

${(\frac{\partial z}{\partial θ})}_{y} = 2 r^{2} \sin 2 θ$

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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.
${(\frac{\partial z}{\partial θ})}_{y}$ .

Short Answer

Step by step solution

Given data

Partial differentiation

Calculation

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Ifz=x2+2y2,x=rcosθ,y=rsinθfind the following partial derivatives.(∂z∂θ)y.

Short Answer

Step by step solution

Given data

Partial differentiation

Calculation

One App. One Place for Learning.

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.
${(\frac{\partial z}{\partial θ})}_{y}$ .