Question

Let φ(x, y) = arctan(y/3x). What is the gradient of φ with respect to x?

a) 3/(x² + (y/3)²)
b) -3y/(x² + (y/3)²)
c) 1/(3x² + y²)
d) -y/(3x² + y²)

Answer 1

The gradient of the function φ(x, y) = arctan(y/3x) with respect to x is φ'_x(x, y) = -y/(3x² + y²), which corresponds to option d) -y/(3x² + y²).

Step-by-step explanation:

The student has asked for the gradient of the function φ(x, y) = arctan(y/3x) with respect to x. To find this gradient, we need to compute the partial derivative of the function with respect to x.

Firstly, recall that for a function arctan(u), the derivative with respect to u is given by 1/(1 + u²). In our case, u = y/3x, so when we differentiate φ(x, y) with respect to x, we treat y as a constant. Thus, the partial derivative is:

φ'_x(x, y) = −(1/(1 + (y/3x)²))(y/3x²) = −y/(3x)(9x² + y²) = −y/(9x³ + y²x)

When you simplify the fraction, the denominator becomes 9x² + y², given that the x² terms can be combined. So the simplified form of the partial derivative is:

φ'_x(x, y) = −y/(3x² + y²)

Thus, the correct option for the gradient of φ with respect to x is d) -y/(3x² + y²).