Question

First and second derivative of arctan(x/y) with respect to x and y:

a. 1/(1 + (x/y)²) and -2xy/(x²+y²)²
b. 1/(1 + x²+y²) and -2xy/(x²+y²)²
c. 1/(1 + x²+y²) and 2xy/(x²+y²)²
d. 1/(1 + (x/y)²) and 2xy/(x²+y²)²

Answer 1

The first and second derivatives of arctan(x/y) with respect to x and y are 1/(1 + (x/y)²) and -2xy/(x²+y²)², employing the chain rule for differentiation.

Step-by-step explanation:

The first and second derivatives of arctan(x/y) with respect to x and y are evaluated using the chain rule and the partial differentiation technique. The correct answers are:

• First derivative with respect to x: 1/(1 + (x/y)²)
• Second derivative with respect to y: -2xy/(x²+y²)²

To calculate the first derivative with respect to x, we use the chain rule and treat y as a constant. For the second derivative with respect to y, we differentiate the result from the first derivative with respect to y again using the chain rule, taking into account that both x and y can change.