Calculus Examples

$4 x^{2} - x - \frac{1}{\sqrt{x}}$

Step 1

By the Sum Rule, the derivative of $4 x^{2} - x - \frac{1}{\sqrt{x}}$ with respect to $x$ is $\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- x] + \frac{d}{d x} [- \frac{1}{\sqrt{x}}]$ .

$\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- x] + \frac{d}{d x} [- \frac{1}{\sqrt{x}}]$

Step 2

Evaluate $\frac{d}{d x} [4 x^{2}]$ .

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Step 2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{2}$ with respect to $x$ is $4 \frac{d}{d x} [x^{2}]$ .

$4 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- x] + \frac{d}{d x} [- \frac{1}{\sqrt{x}}]$

Step 2.2

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

$4 (2 x) + \frac{d}{d x} [- x] + \frac{d}{d x} [- \frac{1}{\sqrt{x}}]$

Step 2.3

Multiply $2$ by $4$ .

$8 x + \frac{d}{d x} [- x] + \frac{d}{d x} [- \frac{1}{\sqrt{x}}]$

Step 3

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

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Step 3.1

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

$8 x - \frac{d}{d x} [x] + \frac{d}{d x} [- \frac{1}{\sqrt{x}}]$

Step 3.2

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

$8 x - 1 \cdot 1 + \frac{d}{d x} [- \frac{1}{\sqrt{x}}]$

Step 3.3

Multiply $- 1$ by $1$ .

$8 x - 1 + \frac{d}{d x} [- \frac{1}{\sqrt{x}}]$

Step 4

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

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Step 4.1

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

$8 x - 1 + \frac{d}{d x} [- \frac{1}{x^{\frac{1}{2}}}]$

Step 4.2

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

$8 x - 1 - \frac{d}{d x} [\frac{1}{x^{\frac{1}{2}}}] + \frac{1}{x^{\frac{1}{2}}} \frac{d}{d x} [- 1]$

Step 4.3

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

$8 x - 1 - \frac{d}{d x} [{(x^{\frac{1}{2}})}^{- 1}] + \frac{1}{x^{\frac{1}{2}}} \frac{d}{d x} [- 1]$

Step 4.4

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

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Step 4.4.1

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

$8 x - 1 - (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{\frac{1}{2}}]) + \frac{1}{x^{\frac{1}{2}}} \frac{d}{d x} [- 1]$

Step 4.4.2

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

$8 x - 1 - (- u^{- 2} \frac{d}{d x} [x^{\frac{1}{2}}]) + \frac{1}{x^{\frac{1}{2}}} \frac{d}{d x} [- 1]$

Step 4.4.3

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

$8 x - 1 - (- {(x^{\frac{1}{2}})}^{- 2} \frac{d}{d x} [x^{\frac{1}{2}}]) + \frac{1}{x^{\frac{1}{2}}} \frac{d}{d x} [- 1]$

Step 4.5

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

$8 x - 1 - (- {(x^{\frac{1}{2}})}^{- 2} (\frac{1}{2} x^{\frac{1}{2} - 1})) + \frac{1}{x^{\frac{1}{2}}} \frac{d}{d x} [- 1]$

Step 4.6

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

$8 x - 1 - (- {(x^{\frac{1}{2}})}^{- 2} (\frac{1}{2} x^{\frac{1}{2} - 1})) + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 4.7

Multiply the exponents in ${(x^{\frac{1}{2}})}^{- 2}$ .

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Step 4.7.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$8 x - 1 - (- x^{\frac{1}{2} \cdot - 2} (\frac{1}{2} x^{\frac{1}{2} - 1})) + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 4.7.2

Cancel the common factor of $2$ .

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Step 4.7.2.1

Factor $2$ out of $- 2$ .

$8 x - 1 - (- x^{\frac{1}{2} \cdot (2 (- 1))} (\frac{1}{2} x^{\frac{1}{2} - 1})) + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 4.7.2.2

Cancel the common factor.

$8 x - 1 - (- x^{\frac{1}{2} \cdot (2 \cdot - 1)} (\frac{1}{2} x^{\frac{1}{2} - 1})) + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 4.7.2.3

Rewrite the expression.

$8 x - 1 - (- x^{- 1} (\frac{1}{2} x^{\frac{1}{2} - 1})) + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 4.8

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

$8 x - 1 - (- x^{- 1} (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})) + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 4.9

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

$8 x - 1 - (- x^{- 1} (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})) + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 4.10

Combine the numerators over the common denominator.

$8 x - 1 - (- x^{- 1} (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})) + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 4.11

Simplify the numerator.

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Step 4.11.1

Multiply $- 1$ by $2$ .

$8 x - 1 - (- x^{- 1} (\frac{1}{2} x^{\frac{1 - 2}{2}})) + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 4.11.2

Subtract $2$ from $1$ .

$8 x - 1 - (- x^{- 1} (\frac{1}{2} x^{\frac{- 1}{2}})) + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 4.12

Move the negative in front of the fraction.

$8 x - 1 - (- x^{- 1} (\frac{1}{2} x^{- \frac{1}{2}})) + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 4.13

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

$8 x - 1 - (- x^{- 1} \frac{x^{- \frac{1}{2}}}{2}) + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 4.14

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

$8 x - 1 - - \frac{x^{- \frac{1}{2}} x^{- 1}}{2} + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 4.15

Multiply $x^{- \frac{1}{2}}$ by $x^{- 1}$ by adding the exponents.

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Step 4.15.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$8 x - 1 - - \frac{x^{- \frac{1}{2} - 1}}{2} + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 4.15.2

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

$8 x - 1 - - \frac{x^{- \frac{1}{2} - 1 \cdot \frac{2}{2}}}{2} + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 4.15.3

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

$8 x - 1 - - \frac{x^{- \frac{1}{2} + \frac{- 1 \cdot 2}{2}}}{2} + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 4.15.4

Combine the numerators over the common denominator.

$8 x - 1 - - \frac{x^{\frac{- 1 - 1 \cdot 2}{2}}}{2} + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 4.15.5

Simplify the numerator.

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Step 4.15.5.1

Multiply $- 1$ by $2$ .

$8 x - 1 - - \frac{x^{\frac{- 1 - 2}{2}}}{2} + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 4.15.5.2

Subtract $2$ from $- 1$ .

$8 x - 1 - - \frac{x^{\frac{- 3}{2}}}{2} + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 4.15.6

Move the negative in front of the fraction.

$8 x - 1 - - \frac{x^{- \frac{3}{2}}}{2} + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 4.16

Move $x^{- \frac{3}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$8 x - 1 - - \frac{1}{2 x^{\frac{3}{2}}} + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 4.17

Multiply $- 1$ by $- 1$ .

$8 x - 1 + 1 \frac{1}{2 x^{\frac{3}{2}}} + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 4.18

Multiply $\frac{1}{2 x^{\frac{3}{2}}}$ by $1$ .

$8 x - 1 + \frac{1}{2 x^{\frac{3}{2}}} + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 4.19

Multiply $\frac{1}{x^{\frac{1}{2}}}$ by $0$ .

$8 x - 1 + \frac{1}{2 x^{\frac{3}{2}}} + 0$

Step 4.20

Add $\frac{1}{2 x^{\frac{3}{2}}}$ and $0$ .

$8 x - 1 + \frac{1}{2 x^{\frac{3}{2}}}$