Chapter 4: Q17P (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.
${(\frac{\partial z}{\partial r})}_{x}$

Short Answer

Expert verified

The value of provided equation is $(\frac{\partial z}{\partial r}), = 4 r$ .

Step by step solution

Given data

The given data are listed below

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

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

Square the equation 2 and 3, then add.

$x^{2} + y^{2} = r^{2}$ ...(4)

From equation (1),

$z = x^{2} + 2 (x^{2} + y^{2}) - 2 x^{2} z = - x^{2} + 2 r^{2}$

Now, differentiate partially with respect to ,

${(\frac{\partial z}{\partial r})}_{x} = \frac{\partial}{\partial r} [- x^{2} + 2 r^{2}] {(\frac{\partial z}{\partial r})}_{x} = 4 r$

Hence, ${(\frac{\partial z}{\partial r})}_{x} = 4 r$

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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.
${(\frac{\partial z}{\partial r})}_{x}$

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