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

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