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Chapter 4: Q8P (page 191)

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

Short Answer

Expert verified

The value of provided equation is, $(\frac{鈭� z}{鈭� x}) = - 2 x$ .

Step by step solution

01

Step 1: Given data

The given data are listed below

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

02

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.

03

Calculation

Square the equation 2 and 3, then add.

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

From equation (1),

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

Take the derivative ofwith respect to ,

role="math" localid="1659095459640" ${(\frac{鈭� z}{鈭� x})}_{r} = \frac{鈭�}{鈭� x} (- x^{2} + 2 r^{2}) {(\frac{鈭� z}{鈭� x})}_{r} = - 2 x$

Therefore, ${(\frac{鈭� z}{鈭� x})}_{r} = - 2 x$ .

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