Question

How do I solve this question

Answer 1

The correct answer choice is D) 2.

Explanation:

To find the derivative of y with respect to x (dy/dx), we'll first need to use the chain rule for differentiation. The chain rule states that the derivative of a composite function is the derivative of the outer function multiplied by the derivative of the inner function.

Here, our function is y = ln(x^2 + y^2). Let's take the derivative with respect to x:

dy/dx = d/dx[ln(x^2 + y^2)]

= 1/(x^2 + y^2) * d/dx[x^2 + y^2]

= 1/(x^2 + y^2) * (2x + 2y * dy/dx)

We need to solve for dy/dx, so let's rearrange the equation:

dy/dx * (1 - 2y/(x^2 + y^2)) = 2x/(x^2 + y^2)

dy/dx = 2x/(x^2 + y^2 + 2y)

Now, let's substitute the point (1,0) into the equation:

dy/dx = 2*1/(1^2 + 0^2 + 2*0)

= 2/1

= 2

So, the value of dy/dx at the point (1, 0) is 2.

Answer 2

D) 2

Explanation:

Let's find the derivative
`\sf (dy)/(dx)` with respect to
`\sf x` for the given function
`\sf y = \ln(x^2 + y^2)`.

Since the expression involves both
`\sf x` and
`\sf y`, we'll use implicit differentiation.

`\sf y = \ln(x^2 + y^2)`

Differentiate both sides with respect to
`\sf x`:

`\sf (dy)/(dx) = (1)/(x^2 + y^2) \cdot (2x + 2y \cdot (dy)/(dx))`

Now, we are interested in finding
`\sf (dy)/(dx)` at the point
`\sf (1, 0)`.

Substitute
`\sf x = 1` and
`\sf y = 0` into the expression:

`\sf (dy)/(dx) = (1)/(1^2 + 0^2) \cdot (2 \cdot 1 + 2 \cdot 0 \cdot (dy)/(dx))`

`\sf (dy)/(dx) = (2)/(1)`

`\sf (dy)/(dx) = 2`

So, the value of
`\sf (dy)/(dx)` at the point
`\sf (1, 0)` is
`\sf 2`.