Calculus Examples

$y = 7 \arctan (x - \sqrt{1 + x^{2}})$

Step 1

Differentiate using the Constant Multiple Rule.

Tap for more steps...

Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1 + x^{2}}$ as ${(1 + x^{2})}^{\frac{1}{2}}$ .

$\frac{d}{d x} [7 \arctan (x - {(1 + x^{2})}^{\frac{1}{2}})]$

Step 1.2

Since $7$ is constant with respect to $x$ , the derivative of $7 \arctan (x - {(1 + x^{2})}^{\frac{1}{2}})$ with respect to $x$ is $7 \frac{d}{d x} [\arctan (x - {(1 + x^{2})}^{\frac{1}{2}})]$ .

$7 \frac{d}{d x} [\arctan (x - {(1 + x^{2})}^{\frac{1}{2}})]$

Step 2

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 $g (x) = x - {(1 + x^{2})}^{\frac{1}{2}}$ .

Tap for more steps...

Step 2.1

To apply the Chain Rule, set $u_{1}$ as $x - {(1 + x^{2})}^{\frac{1}{2}}$ .

$7 (\frac{d}{d u_{1}} [\arctan (u_{1})] \frac{d}{d x} [x - {(1 + x^{2})}^{\frac{1}{2}}])$

Step 2.2

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

$7 (\frac{1}{1 + {u_{1}}^{2}} \frac{d}{d x} [x - {(1 + x^{2})}^{\frac{1}{2}}])$

Step 2.3

Replace all occurrences of $u_{1}$ with $x - {(1 + x^{2})}^{\frac{1}{2}}$ .

$7 (\frac{1}{1 + {(x - {(1 + x^{2})}^{\frac{1}{2}})}^{2}} \frac{d}{d x} [x - {(1 + x^{2})}^{\frac{1}{2}}])$

Step 3

Differentiate.

Tap for more steps...

Step 3.1

Combine $\frac{1}{1 + {(x - {(1 + x^{2})}^{\frac{1}{2}})}^{2}}$ and $7$ .

$\frac{7}{1 + {(x - {(1 + x^{2})}^{\frac{1}{2}})}^{2}} \frac{d}{d x} [x - {(1 + x^{2})}^{\frac{1}{2}}]$

Step 3.2

By the Sum Rule, the derivative of $x - {(1 + x^{2})}^{\frac{1}{2}}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- {(1 + x^{2})}^{\frac{1}{2}}]$ .

$\frac{7}{1 + {(x - {(1 + x^{2})}^{\frac{1}{2}})}^{2}} (\frac{d}{d x} [x] + \frac{d}{d x} [- {(1 + x^{2})}^{\frac{1}{2}}])$

Step 3.3

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

$\frac{7}{1 + {(x - {(1 + x^{2})}^{\frac{1}{2}})}^{2}} (1 + \frac{d}{d x} [- {(1 + x^{2})}^{\frac{1}{2}}])$

Step 3.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- {(1 + x^{2})}^{\frac{1}{2}}$ with respect to $x$ is $- \frac{d}{d x} [{(1 + x^{2})}^{\frac{1}{2}}]$ .

$\frac{7}{1 + {(x - {(1 + x^{2})}^{\frac{1}{2}})}^{2}} (1 - \frac{d}{d x} [{(1 + x^{2})}^{\frac{1}{2}}])$

Step 4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = 1 + x^{2}$ .

Tap for more steps...

Step 4.1

To apply the Chain Rule, set $u_{2}$ as $1 + x^{2}$ .

$\frac{7}{1 + {(x - {(1 + x^{2})}^{\frac{1}{2}})}^{2}} (1 - (\frac{d}{d u_{2}} [{u_{2}}^{\frac{1}{2}}] \frac{d}{d x} [1 + x^{2}]))$

Step 4.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = \frac{1}{2}$ .

$\frac{7}{1 + {(x - {(1 + x^{2})}^{\frac{1}{2}})}^{2}} (1 - (\frac{1}{2} {u_{2}}^{\frac{1}{2} - 1} \frac{d}{d x} [1 + x^{2}]))$

Step 4.3

Replace all occurrences of $u_{2}$ with $1 + x^{2}$ .

$\frac{7}{1 + {(x - {(1 + x^{2})}^{\frac{1}{2}})}^{2}} (1 - (\frac{1}{2} {(1 + x^{2})}^{\frac{1}{2} - 1} \frac{d}{d x} [1 + x^{2}]))$

Step 5

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{7}{1 + {(x - {(1 + x^{2})}^{\frac{1}{2}})}^{2}} (1 - (\frac{1}{2} {(1 + x^{2})}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [1 + x^{2}]))$

Step 6

Combine $- 1$ and $\frac{2}{2}$ .

$\frac{7}{1 + {(x - {(1 + x^{2})}^{\frac{1}{2}})}^{2}} (1 - (\frac{1}{2} {(1 + x^{2})}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [1 + x^{2}]))$

Step 7

Combine the numerators over the common denominator.

$\frac{7}{1 + {(x - {(1 + x^{2})}^{\frac{1}{2}})}^{2}} (1 - (\frac{1}{2} {(1 + x^{2})}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [1 + x^{2}]))$

Step 8

Simplify the numerator.

Tap for more steps...

Step 8.1

Multiply $- 1$ by $2$ .

$\frac{7}{1 + {(x - {(1 + x^{2})}^{\frac{1}{2}})}^{2}} (1 - (\frac{1}{2} {(1 + x^{2})}^{\frac{1 - 2}{2}} \frac{d}{d x} [1 + x^{2}]))$

Step 8.2

Subtract $2$ from $1$ .

$\frac{7}{1 + {(x - {(1 + x^{2})}^{\frac{1}{2}})}^{2}} (1 - (\frac{1}{2} {(1 + x^{2})}^{\frac{- 1}{2}} \frac{d}{d x} [1 + x^{2}]))$

Step 9

Combine fractions.

Tap for more steps...

Step 9.1

Move the negative in front of the fraction.

$\frac{7}{1 + {(x - {(1 + x^{2})}^{\frac{1}{2}})}^{2}} (1 - (\frac{1}{2} {(1 + x^{2})}^{- \frac{1}{2}} \frac{d}{d x} [1 + x^{2}]))$

Step 9.2

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

$\frac{7}{1 + {(x - {(1 + x^{2})}^{\frac{1}{2}})}^{2}} (1 - (\frac{{(1 + x^{2})}^{- \frac{1}{2}}}{2} \frac{d}{d x} [1 + x^{2}]))$

Step 9.3

Move ${(1 + x^{2})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{7}{1 + {(x - {(1 + x^{2})}^{\frac{1}{2}})}^{2}} (1 - (\frac{1}{2 {(1 + x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [1 + x^{2}]))$

Step 10

By the Sum Rule, the derivative of $1 + x^{2}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [x^{2}]$ .

$\frac{7}{1 + {(x - {(1 + x^{2})}^{\frac{1}{2}})}^{2}} (1 - \frac{1}{2 {(1 + x^{2})}^{\frac{1}{2}}} (\frac{d}{d x} [1] + \frac{d}{d x} [x^{2}]))$

Step 11

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

$\frac{7}{1 + {(x - {(1 + x^{2})}^{\frac{1}{2}})}^{2}} (1 - \frac{1}{2 {(1 + x^{2})}^{\frac{1}{2}}} (0 + \frac{d}{d x} [x^{2}]))$

Step 12

Add $0$ and $\frac{d}{d x} [x^{2}]$ .

$\frac{7}{1 + {(x - {(1 + x^{2})}^{\frac{1}{2}})}^{2}} (1 - \frac{1}{2 {(1 + x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [x^{2}])$

Step 13

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$\frac{7}{1 + {(x - {(1 + x^{2})}^{\frac{1}{2}})}^{2}} (1 - \frac{1}{2 {(1 + x^{2})}^{\frac{1}{2}}} (2 x))$

Step 14

Simplify terms.

Tap for more steps...

Step 14.1

Multiply $2$ by $- 1$ .

$\frac{7}{1 + {(x - {(1 + x^{2})}^{\frac{1}{2}})}^{2}} (1 - 2 \frac{1}{2 {(1 + x^{2})}^{\frac{1}{2}}} x)$

Step 14.2

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

$\frac{7}{1 + {(x - {(1 + x^{2})}^{\frac{1}{2}})}^{2}} (1 + \frac{- 2}{2 {(1 + x^{2})}^{\frac{1}{2}}} x)$

Step 14.3

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

$\frac{7}{1 + {(x - {(1 + x^{2})}^{\frac{1}{2}})}^{2}} (1 + \frac{- 2 x}{2 {(1 + x^{2})}^{\frac{1}{2}}})$

Step 14.4

Factor $2$ out of $- 2 x$ .

$\frac{7}{1 + {(x - {(1 + x^{2})}^{\frac{1}{2}})}^{2}} (1 + \frac{2 (- x)}{2 {(1 + x^{2})}^{\frac{1}{2}}})$

Step 15

Cancel the common factors.

Tap for more steps...

Step 15.1

Factor $2$ out of $2 {(1 + x^{2})}^{\frac{1}{2}}$ .

$\frac{7}{1 + {(x - {(1 + x^{2})}^{\frac{1}{2}})}^{2}} (1 + \frac{2 (- x)}{2 ({(1 + x^{2})}^{\frac{1}{2}})})$

Step 15.2

Cancel the common factor.

$\frac{7}{1 + {(x - {(1 + x^{2})}^{\frac{1}{2}})}^{2}} (1 + \frac{2 (- x)}{2 {(1 + x^{2})}^{\frac{1}{2}}})$

Step 15.3

Rewrite the expression.

$\frac{7}{1 + {(x - {(1 + x^{2})}^{\frac{1}{2}})}^{2}} (1 + \frac{- x}{{(1 + x^{2})}^{\frac{1}{2}}})$

Step 16

Move the negative in front of the fraction.

$\frac{7}{1 + {(x - {(1 + x^{2})}^{\frac{1}{2}})}^{2}} (1 - \frac{x}{{(1 + x^{2})}^{\frac{1}{2}}})$

Step 17

Reorder the factors of $\frac{7}{1 + {(x - {(1 + x^{2})}^{\frac{1}{2}})}^{2}} (1 - \frac{x}{{(1 + x^{2})}^{\frac{1}{2}}})$ .

$(1 - \frac{x}{{(1 + x^{2})}^{\frac{1}{2}}}) \frac{7}{1 + {(x - {(1 + x^{2})}^{\frac{1}{2}})}^{2}}$