Chapter 4: Q9P (page 191)

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

Short Answer

Expert verified

The value of provided equation is ${(\frac{\partial z}{\partial x})}_{θ} = 2 x (1 + 2 \tan^{2} θ)$ .

Step by step solution

Given data

The given data are listed below

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

Step 2: 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,

$\frac{y}{x} = \tan θ y = x \tan θ$

Substitute the value of y in equation 1.

$z = x^{2} + 2 {(x \tan θ)}^{2} z = x^{2} (1 + 2 \tan^{2} θ)$

Take the partial derivative of zwith respect to x,

role="math" localid="1659009652553" ${(\frac{\partial z}{\partial x})}_{θ} = \frac{\partial}{\partial x} (x^{2} (1 + 2 \tan^{2} θ)) {(\frac{\partial z}{\partial x})}_{θ} = 2 x (1 + 2 \tan^{2} θ)$

Therefore, ${(\frac{\partial z}{\partial x})}_{θ} = 2 x (1 + 2 \tan^{2} θ)$ .

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

Short Answer

Step by step solution

Given data

Step 2: Partial differentiation

Calculation

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if z=x2+2y2,x=rcosθ,y=sinθ, find the following partial derivatives.(∂z∂x)θ.

Short Answer

Step by step solution

Given data

Step 2: Partial differentiation

Calculation

One App. One Place for Learning.

Most popular questions from this chapter

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