Chapter 4: Q13P (page 191)

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

Short Answer

Expert verified

The value of provide equation is ${(\frac{\partial z}{\partial θ})}_{x} = 4 r^{2} \sin θ \cos θ$ .

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 3,

$z = x^{2} + 2 {(r \sin θ)}^{2} z = x^{2} + 2 r^{2} \sin^{2} θ$

Now, take the partial derivative with respect to in above equation,

role="math" localid="1659091215113" ${(\frac{\partial z}{\partial θ})}_{x} = \frac{\partial}{\partial θ} (x^{2} + 2 r^{2} \sin^{2} θ) {(\frac{\partial z}{\partial θ})}_{x} = 4 r^{2} \sin θ \cos θ$

Hence, ${(\frac{\partial z}{\partial θ})}_{x} = 4 r^{2} \sin θ \cos θ$

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

Short Answer

Step by step solution

Given data

Partial differentiation

Calculation

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

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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Study anywhere. Anytime. Across all devices.

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