Find the Derivative – d/dx arctan(xy)

$\arctan (x y)$

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

Tap for more steps…

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

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

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

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

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

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

Differentiate.

Tap for more steps…

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

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

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

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

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

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

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

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

Simplify.

Tap for more steps…

Apply the product rule to $x y$ .

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

Reorder terms.

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

Find the Derivative – d/dx arctan(xy)

Download our
App from the store

Create a High Performed UI/UX Design from a Silicon Valley.

Download our App from the store

Create a High Performed UI/UX Design from a Silicon Valley.

Download our
App from the store