Calculus Examples

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

Step 1

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

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

Step 2

Evaluate $\frac{d}{d x} [2 x]$ .

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Step 2.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

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

Step 2.2

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

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

Step 2.3

Multiply $2$ by $1$ .

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

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

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

Step 3.9

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

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

Step 3.10

Combine the numerators over the common denominator.

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

Step 3.11

Simplify the numerator.

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Step 3.11.1

Multiply $- 1$ by $2$ .

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

Step 3.11.2

Subtract $2$ from $1$ .

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

Step 3.12

Move the negative in front of the fraction.

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

Step 3.13

Add $2 x + 1$ and $0$ .

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

Step 3.14

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

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

Step 3.15

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

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

Step 4

Reorder terms.

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