Calculus Examples

$\arctan (\frac{x + y}{1 - x y})$

Step 1

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 $g (y) = \frac{x + y}{1 - x y}$ .

Tap for more steps...

Step 1.1

To apply the Chain Rule, set $u$ as $\frac{x + y}{1 - x y}$ .

$\frac{d}{d u} [\arctan (u)] \frac{d}{d y} [\frac{x + y}{1 - x y}]$

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u$ with $\frac{x + y}{1 - x y}$ .

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

Step 2

Differentiate using the Quotient Rule which states that $\frac{d}{d y} [\frac{f (y)}{g (y)}]$ is $\frac{g (y) \frac{d}{d y} [f (y)] - f (y) \frac{d}{d y} [g (y)]}{{g (y)}^{2}}$ where $f (y) = x + y$ and $g (y) = 1 - x y$ .

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

Step 3

Differentiate.

Tap for more steps...

Step 3.1

By the Sum Rule, the derivative of $x + y$ with respect to $y$ is $\frac{d}{d y} [x] + \frac{d}{d y} [y]$ .

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

Step 3.2

Since $x$ is constant with respect to $y$ , the derivative of $x$ with respect to $y$ is $0$ .

$\frac{1}{1 + {(\frac{x + y}{1 - x y})}^{2}} \cdot \frac{(1 - x y) (0 + \frac{d}{d y} [y]) - (x + y) \frac{d}{d y} [1 - x y]}{{(1 - x y)}^{2}}$

Step 3.3

Add $0$ and $\frac{d}{d y} [y]$ .

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

Step 3.4

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

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

Step 3.5

Multiply $1 - x y$ by $1$ .

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

Step 3.6

By the Sum Rule, the derivative of $1 - x y$ with respect to $y$ is $\frac{d}{d y} [1] + \frac{d}{d y} [- x y]$ .

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

Step 3.7

Since $1$ is constant with respect to $y$ , the derivative of $1$ with respect to $y$ is $0$ .

$\frac{1}{1 + {(\frac{x + y}{1 - x y})}^{2}} \cdot \frac{1 - x y - (x + y) (0 + \frac{d}{d y} [- x y])}{{(1 - x y)}^{2}}$

Step 3.8

Add $0$ and $\frac{d}{d y} [- x y]$ .

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

Step 3.9

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

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

Step 3.10

Multiply.

Tap for more steps...

Step 3.10.1

Multiply $- 1$ by $- 1$ .

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

Step 3.10.2

Multiply $x + y$ by $1$ .

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

Step 3.11

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

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

Step 3.12

Combine fractions.

Tap for more steps...

Step 3.12.1

Multiply $x + y$ by $1$ .

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

Step 3.12.2

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

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

Step 4

Simplify.

Tap for more steps...

Step 4.1

Apply the product rule to $\frac{x + y}{1 - x y}$ .

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Simplify the numerator.

Tap for more steps...

Step 4.3.1

Combine the opposite terms in $1 - x y + x \cdot x + x y$ .

Tap for more steps...

Step 4.3.1.1

Add $- x y$ and $x y$ .

$\frac{1 + x \cdot x + 0}{(1 + \frac{{(x + y)}^{2}}{{(1 - x y)}^{2}}) {(1 - x y)}^{2}}$

Step 4.3.1.2

Add $1 + x \cdot x$ and $0$ .

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

Step 4.3.2

Multiply $x$ by $x$ .

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

Step 4.4

Combine terms.

Tap for more steps...

Step 4.4.1

Write $1$ as a fraction with a common denominator.

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

Step 4.4.2

Combine the numerators over the common denominator.

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

Step 4.4.3

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

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

Step 4.4.4

Cancel the common factor of ${(1 - x y)}^{2}$ .

Tap for more steps...

Step 4.4.4.1

Cancel the common factor.

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

Step 4.4.4.2

Divide ${(1 - x y)}^{2} + {(x + y)}^{2}$ by $1$ .

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

Step 4.5

Reorder terms.

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

Step 4.6

Simplify the denominator.

Tap for more steps...

Step 4.6.1

Rewrite ${(1 - x y)}^{2}$ as $(1 - x y) (1 - x y)$ .

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

Step 4.6.2

Expand $(1 - x y) (1 - x y)$ using the FOIL Method.

Tap for more steps...

Step 4.6.2.1

Apply the distributive property.

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

Step 4.6.2.2

Apply the distributive property.

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

Step 4.6.2.3

Apply the distributive property.

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

Step 4.6.3

Simplify and combine like terms.

Tap for more steps...

Step 4.6.3.1

Simplify each term.

Tap for more steps...

Step 4.6.3.1.1

Multiply $1$ by $1$ .

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

Step 4.6.3.1.2

Multiply $- x y$ by $1$ .

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

Step 4.6.3.1.3

Multiply $- 1$ by $1$ .

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

Step 4.6.3.1.4

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 4.6.3.1.4.1

Move $x$ .

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

Step 4.6.3.1.4.2

Multiply $x$ by $x$ .

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

Step 4.6.3.1.5

Multiply $y$ by $y$ by adding the exponents.

Tap for more steps...

Step 4.6.3.1.5.1

Move $y$ .

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

Step 4.6.3.1.5.2

Multiply $y$ by $y$ .

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

Step 4.6.3.1.6

Multiply $- x^{2} y^{2} \cdot - 1$ .

Tap for more steps...

Step 4.6.3.1.6.1

Multiply $- 1$ by $- 1$ .

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

Step 4.6.3.1.6.2

Multiply $x^{2}$ by $1$ .

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

Step 4.6.3.2

Subtract $x y$ from $- x y$ .

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

Step 4.6.4

Rewrite ${(x + y)}^{2}$ as $(x + y) (x + y)$ .

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

Step 4.6.5

Expand $(x + y) (x + y)$ using the FOIL Method.

Tap for more steps...

Step 4.6.5.1

Apply the distributive property.

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

Step 4.6.5.2

Apply the distributive property.

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

Step 4.6.5.3

Apply the distributive property.

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

Step 4.6.6

Simplify and combine like terms.

Tap for more steps...

Step 4.6.6.1

Simplify each term.

Tap for more steps...

Step 4.6.6.1.1

Multiply $x$ by $x$ .

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

Step 4.6.6.1.2

Multiply $y$ by $y$ .

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

Step 4.6.6.2

Add $x y$ and $y x$ .

Tap for more steps...

Step 4.6.6.2.1

Reorder $y$ and $x$ .

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

Step 4.6.6.2.2

Add $x y$ and $x y$ .

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

Step 4.6.7

Add $- 2 x y$ and $2 x y$ .

$\frac{x^{2} + 1}{1 + x^{2} y^{2} + x^{2} + 0 + y^{2}}$

Step 4.6.8

Add $1$ and $0$ .

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

Step 4.6.9

Rewrite $x^{2} y^{2} + x^{2} + 1 + y^{2}$ in a factored form.

Tap for more steps...

Step 4.6.9.1

Factor out the greatest common factor from each group.

Tap for more steps...

Step 4.6.9.1.1

Group the first two terms and the last two terms.

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

Step 4.6.9.1.2

Factor out the greatest common factor (GCF) from each group.

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

Step 4.6.9.2

Factor the polynomial by factoring out the greatest common factor, $y^{2} + 1$ .

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

Step 4.7

Cancel the common factor of $x^{2} + 1$ .

Tap for more steps...

Step 4.7.1

Cancel the common factor.

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

Step 4.7.2

Rewrite the expression.

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