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Chapter 13: Problem 24

Find the first partial derivatives of the following functions. $$f(x, y)=\sqrt{x^{2} y^{3}}$$

Short Answer

Expert verified
Answer: The first partial derivatives of the function are: $$\frac{\partial f}{\partial x} = \frac{y}{\sqrt{x}}$$ $$\frac{\partial f}{\partial y} = \frac{3x}{2\sqrt{y}}$$

Step by step solution

01

Rewrite function as a power

Rewrite the function in power notation, it will make differentiation easier: $$f(x, y) = (x^{2} y^{3})^{\frac{1}{2}}$$
02

Differentiate with respect to x

Using the chain rule: $$\frac{\partial f}{\partial x} = \frac{1}{2}(x^{2} y^{3})^{-\frac{1}{2}} \cdot \frac{\partial}{\partial x} (x^{2} y^{3})$$ Now we need to differentiate the term inside the parentheses with respect to x: $$\frac{\partial}{\partial x}(x^{2} y^{3}) = 2x y^{3}$$
03

Substitute back into the formula

Substitute the differentiated term back into the formula for the partial derivative: $$\frac{\partial f}{\partial x} = \frac{1}{2}(x^{2} y^{3})^{-\frac{1}{2}} \cdot (2xy^{3})$$
04

Simplify the expression

Simplify the expression to obtain our final result: $$\frac{\partial f}{\partial x} = \frac{2xy^{3}}{2\sqrt{x^{2}y^{3}}} = \frac{y}{\sqrt{x}}$$ Next, we will find the partial derivative of f with respect to y, denoted as \(\frac{\partial f}{\partial y}\).
05

Differentiate with respect to y

Using the chain rule again: $$\frac{\partial f}{\partial y} = \frac{1}{2}(x^{2} y^{3})^{-\frac{1}{2}} \cdot \frac{\partial}{\partial y} (x^{2} y^{3})$$ We need to differentiate the term inside the parentheses with respect to y: $$\frac{\partial}{\partial y}(x^{2} y^{3}) = 3x^{2} y^{2}$$
06

Substitute back into the formula

Substitute the differentiated term back into the formula for the partial derivative: $$\frac{\partial f}{\partial y} = \frac{1}{2}(x^{2} y^{3})^{-\frac{1}{2}} \cdot (3x^{2}y^{2})$$
07

Simplify the expression

Simplify the expression to obtain our final result: $$\frac{\partial f}{\partial y} = \frac{3x^{2}y^{2}}{2\sqrt{x^{2}y^{3}}} = \frac{3x}{2\sqrt{y}}$$ So, the first partial derivatives of the function are: $$\frac{\partial f}{\partial x} = \frac{y}{\sqrt{x}}$$ $$\frac{\partial f}{\partial y} = \frac{3x}{2\sqrt{y}}$$

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