Find the Derivative - d/dy arctan(xy)

$\arctan (x y)$

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = \arctan (y)$ and $g (y) = x y$ .

To apply the Chain Rule, set $u$ as $x y$ .

$\frac{d}{d u} [\arctan (u)] \frac{d}{d y} [x y]$

The derivative of $\arctan (u)$ with respect to $u$ is $\frac{1}{1 + u^{2}}$ .

$\frac{1}{1 + u^{2}} \frac{d}{d y} [x y]$

Replace all occurrences of $u$ with $x y$ .

$\frac{1}{1 + {(x y)}^{2}} \frac{d}{d y} [x y]$

Differentiate.

Since $x$ is constant with respect to $y$ , the derivative of $x y$ with respect to $y$ is $x \frac{d}{d y} [y]$ .

$\frac{1}{1 + {(x y)}^{2}} (x \frac{d}{d y} [y])$

Combine $x$ and $\frac{1}{1 + {(x y)}^{2}}$ .

$\frac{x}{1 + {(x y)}^{2}} \frac{d}{d y} [y]$

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 1$ .

$\frac{x}{1 + {(x y)}^{2}} \cdot 1$

Multiply $\frac{x}{1 + {(x y)}^{2}}$ by $1$ .

$\frac{x}{1 + {(x y)}^{2}}$

Simplify.

Apply the product rule to $x y$ .

$\frac{x}{1 + x^{2} y^{2}}$

Reorder terms.

$\frac{x}{x^{2} y^{2} + 1}$

Find the Derivative - d/dy arctan(xy)

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