Question

$$\text { If } u=x^{2} /\left(x^{2}+y^{2}\right), \text { find } \partial u / \partial x, \partial u / \partial y$$.

Short answer

\(\frac{\partial u}{\partial x} = \frac{2xy^2}{(x^2 + y^2)^2}\), \(\frac{\partial u}{\partial y} = \frac{-2x^2 y}{(x^2 + y^2)^2}\)

Step-by-step solution

Understand the Function

Consider the function given: \[u = \frac{x^2}{x^2 + y^2}\]This is a function of two variables, x and y.

Find the Partial Derivative with Respect to x

Use the quotient rule for derivatives, which is \[ \frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2} \]For the function given, identify:\[ f(x, y) = x^2 \]\[ g(x, y) = x^2 + y^2 \]Differentiate each part:\[ f'(x, y) = 2x \]\[ g'(x, y) = 2x \]Then, apply the quotient rule:\[ \frac{\partial u}{\partial x} = \frac{(2x)(x^2 + y^2) - (x^2)(2x)}{(x^2 + y^2)^2} \]Simplify the numerator:\[ = \frac{2x(x^2 + y^2) - 2x^3}{(x^2 + y^2)^2} \]\[ = \frac{2x^3 + 2xy^2 - 2x^3}{(x^2 + y^2)^2} \]\[ = \frac{2xy^2}{(x^2 + y^2)^2} \]

Find the Partial Derivative with Respect to y

Again, use the quotient rule. For this, identify:\[ f(x, y) = x^2 \]\[ g(x, y) = x^2 + y^2 \]Now find the derivatives with respect to y:\[ f'(x, y) = 0 \] (since x is treated as a constant with respect to y)\[ g'(x, y) = 2y \]Apply the quotient rule:\[ \frac{\partial u}{\partial y} = \frac{(0)(x^2 + y^2) - (x^2)(2y)}{(x^2 + y^2)^2} \]\[ = \frac{-2x^2 y}{(x^2 + y^2)^2} \]

Key concepts

multivariable calculus

Multivariable calculus extends the concepts of calculus to functions with more than one variable. This subject helps us understand how several variables interact with each other. For instance, we have the function: \[ u = \frac{x^2}{x^2 + y^2} \] This function depends on two variables, x and y. In multivariable calculus, you not only differentiate and integrate functions of one variable, but you also study how functions vary with changes in multiple variables. The partial derivatives are tools used for this purpose, showing how a function changes as one of its input variables changes, while keeping the other variables constant. This is crucial in fields like physics, engineering, economics, and more.

quotient rule

The quotient rule is a technique for differentiating a function that is the ratio of two other functions. The formula for the quotient rule is: \[ \frac{d}{dx} \frac{f(x)}{g(x)} = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2} \] In the given exercise, we have: \[ u = \frac{x^2}{x^2 + y^2} \] Here, we identify:

• f(x, y) = x^2
• g(x, y) = x^2 + y^2

By using the quotient rule, we first find the derivatives of f and g. Then, the quotient rule combines these derivatives to give the derivative of the entire function. For the partial derivative with respect to x, we apply the quotient rule with respect to x, treating y as a constant: \[ \frac{\frac{\text{\footnotesize v}}{\text{\footnotesize div}}{partial u}{partial x} = \frac{(2x)(x^2 + y^2) - (x^2)(2x)}{(x^2 + y^2)^2 \] After simplification, it becomes: \[ = \frac{2xy^2}{(x^2 + y^2)^2} \]
When taking the partial derivative with respect to y, we again use the quotient rule, this time treating x as a constant: \[ \frac{\frac{\text{\footnotesize v}}{\text{\footnotesize div}}{partial u}\frac{\text{\footnotesize y}\text{\footnotesize \right] - (x^2)(2y}]{fa}{x^2 + y^2 \] After simplification, it becomes: \[ = \frac{-2x^2 y}{(x^2 + y^2)^2} \] This method helps simplify and correctly differentiate complex ratios of functions.

partial differentiation

Partial differentiation focuses on finding the derivative of multivariable functions with respect to one variable, while holding others constant. This is useful in analyzing how changes in one variable affect the overall function. Consider the given function: \[ u = \frac{x^2}{x^2 + y^2} \] To find the partial derivative with respect to x, we treat y as a constant:

• Identify f(x, y) = x^2
• Identify g(x, y) = x^2 + y^2
• Find f'(x): 2x

Next, apply the quotient rule and simplify: \[ \frac{\boolean_x} = \frac{(2x)(x^2 + y^2) - (x^2)(2x)}{(x^2 + y2)^2) = \frac{2xy^2}{(x^2 + y^2)^2} \] Similarly, for the partial derivative with respect to y, treat x as a constant:

• f(x, y) = x^2
• g(x, y) = x^2 + y^2
• f'(x, y) = 0
• g'(y) = 2y

Apply the quotient rule and simplify: \[ \frac{\text{partial u}}{\text{\text{\text{partial y}}combo_neural network}{2}