Use implicit differentiation to find the derivative for: arctan(xy)=1+x

To find the derivative of the equation arctan(xy) = 1 + x using implicit differentiation, we need to differentiate both sides of the equation with respect to x, treating y as a function of x.

Let's go through the steps:

Step 1: Differentiate both sides of the equation with respect to x.
The derivative of arctan(xy) with respect to x can be found using the chain rule. Recall that the derivative of arctan(u) with respect to u is 1/(1+u^2). So, we have:

(d/dx)[arctan(xy)] = d/dx(1 + x)

Step 2: Apply the chain rule to differentiate arctan(xy) with respect to x.
Let's differentiate the left side first. Since we have arctan(xy), we need to use the chain rule.

The chain rule states that if we have a function f(g(x)), then its derivative is f'(g(x)) * g'(x).

So, we apply the chain rule to arctan(xy):

d/dx[arctan(xy)] = 1/(1 + (xy)^2) * d/dx[xy]

Step 3: Differentiate 1 + x with respect to x.
The derivative of 1 + x with respect to x is simply 1.

Combining both sides of the equation, we have:

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

Step 4: Solve for d/dx[xy].
To isolate d/dx[xy], we can multiply both sides of the equation by (1 + (xy)^2):

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

Simplifying the equation, we get:

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

Step 5: Solve for d/dx[xy].
Dividing both sides of the equation by (1 + (xy)^2), we have:

d/dx[xy] = (1 + (xy)^2) / (1 + (xy)^2)

Simplifying the equation further, we get:

d/dx[xy] = 1

So, the derivative of arctan(xy) = 1 + x with respect to x is simply 1.

Therefore, d/dx[arctan(xy)] = 1.