Calculus Examples

$\sqrt{x^{2} + 1} - x$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

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

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

Step 1.1.2

Evaluate $\frac{d}{d x} [\sqrt{x^{2} + 1}]$ .

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

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Step 1.1.2.2.1

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

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

Step 1.1.2.2.2

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

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

Step 1.1.2.2.3

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

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

Step 1.1.2.3

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

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

Step 1.1.2.4

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

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

Step 1.1.2.5

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

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

Step 1.1.2.6

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

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

Step 1.1.2.7

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

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

Step 1.1.2.8

Combine the numerators over the common denominator.

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

Step 1.1.2.9

Simplify the numerator.

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Step 1.1.2.9.1

Multiply $- 1$ by $2$ .

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

Step 1.1.2.9.2

Subtract $2$ from $1$ .

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

Step 1.1.2.10

Move the negative in front of the fraction.

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

Step 1.1.2.11

Add $2 x$ and $0$ .

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

Step 1.1.2.12

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

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

Step 1.1.2.13

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

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

Step 1.1.2.14

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

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

Step 1.1.2.15

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

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

Step 1.1.2.16

Cancel the common factor.

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

Step 1.1.2.17

Rewrite the expression.

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

Step 1.1.3

Evaluate $\frac{d}{d x} [- x]$ .

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Step 1.1.3.1

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

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

Step 1.1.3.2

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

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

Step 1.1.3.3

Multiply $- 1$ by $1$ .

$f' (x) = \frac{x}{{(x^{2} + 1)}^{\frac{1}{2}}} - 1$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{x}{{(x^{2} + 1)}^{\frac{1}{2}}} - 1$ .

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

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{x}{{(x^{2} + 1)}^{\frac{1}{2}}} - 1 = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

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

Step 2.2

Graph each side of the equation. The solution is the x-value of the point of intersection.

No solution

Step 3

Find the values where the derivative is undefined.

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Step 3.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 4

There are no values of $x$ in the domain of the original problem where the derivative is $0$ or undefined.

No critical points found