Chapter 4: Q20P (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.
$\frac{\partial^{2} z}{\partial x \partial θ}$ .

Short Answer

Expert verified

The value of provided equation is $\frac{\partial^{2} z}{\partial x \partial θ} = 8 x \tan θ s e c^{2} θ$

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 2 and 3,

$\frac{y}{x} = \frac{\sin θ}{\cos θ} y = x \tan θ$

Substitute the value of y in equation 1.

role="math" localid="1659088558350" $z = x^{2} + 2 {(x \tan θ)}^{2} z = x^{2} (1 + 2 \tan^{2} θ)$

Take partial derivative of z with respect to x,

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

Take partial derivative of $\frac{\partial z}{\partial x}$ with respect to ,

$\frac{\partial^{2} z}{\partial x \partial θ} = \frac{\partial^{2}}{\partial x \partial θ} 2 x (1 + 2 \tan^{2} θ) \frac{\partial^{2} z}{\partial x \partial θ} = 8 x \tan θ s e c^{2} θ$

Hence,

$\frac{\partial^{2} z}{\partial x \partial θ} = 8 x \tan θ s e c^{2} θ$

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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.
$\frac{\partial^{2} z}{\partial x \partial θ}$ .

Short Answer

Step by step solution

Given data

Partial differentiation

Calculation

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

Short Answer

Step by step solution

Given data

Partial differentiation

Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Circular Motion and Gravitation

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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.
$\frac{\partial^{2} z}{\partial x \partial θ}$ .