Calculus Examples

$y = 2 x^{2} - \sqrt{x} + \frac{5}{x^{2}} + 6$

Step 1

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

$y = 2 x^{2} - x^{\frac{1}{2}} + \frac{5}{x^{2}} + 6$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (2 x^{2} - x^{\frac{1}{2}} + \frac{5}{x^{2}} + 6)$

Step 3

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

By the Sum Rule, the derivative of $2 x^{2} - x^{\frac{1}{2}} + \frac{5}{x^{2}} + 6$ with respect to $x$ is $\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- x^{\frac{1}{2}}] + \frac{d}{d x} [\frac{5}{x^{2}}] + \frac{d}{d x} [6]$ .

$\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- x^{\frac{1}{2}}] + \frac{d}{d x} [\frac{5}{x^{2}}] + \frac{d}{d x} [6]$

Step 4.2

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

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Step 4.2.1

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

$2 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- x^{\frac{1}{2}}] + \frac{d}{d x} [\frac{5}{x^{2}}] + \frac{d}{d x} [6]$

Step 4.2.2

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

$2 (2 x) + \frac{d}{d x} [- x^{\frac{1}{2}}] + \frac{d}{d x} [\frac{5}{x^{2}}] + \frac{d}{d x} [6]$

Step 4.2.3

Multiply $2$ by $2$ .

$4 x + \frac{d}{d x} [- x^{\frac{1}{2}}] + \frac{d}{d x} [\frac{5}{x^{2}}] + \frac{d}{d x} [6]$

Step 4.3

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

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Step 4.3.1

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

$4 x - \frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [\frac{5}{x^{2}}] + \frac{d}{d x} [6]$

Step 4.3.2

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

$4 x - (\frac{1}{2} x^{\frac{1}{2} - 1}) + \frac{d}{d x} [\frac{5}{x^{2}}] + \frac{d}{d x} [6]$

Step 4.3.3

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

$4 x - (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [\frac{5}{x^{2}}] + \frac{d}{d x} [6]$

Step 4.3.4

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

$4 x - (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [\frac{5}{x^{2}}] + \frac{d}{d x} [6]$

Step 4.3.5

Combine the numerators over the common denominator.

$4 x - (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}) + \frac{d}{d x} [\frac{5}{x^{2}}] + \frac{d}{d x} [6]$

Step 4.3.6

Simplify the numerator.

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Step 4.3.6.1

Multiply $- 1$ by $2$ .

$4 x - (\frac{1}{2} x^{\frac{1 - 2}{2}}) + \frac{d}{d x} [\frac{5}{x^{2}}] + \frac{d}{d x} [6]$

Step 4.3.6.2

Subtract $2$ from $1$ .

$4 x - (\frac{1}{2} x^{\frac{- 1}{2}}) + \frac{d}{d x} [\frac{5}{x^{2}}] + \frac{d}{d x} [6]$

Step 4.3.7

Move the negative in front of the fraction.

$4 x - (\frac{1}{2} x^{- \frac{1}{2}}) + \frac{d}{d x} [\frac{5}{x^{2}}] + \frac{d}{d x} [6]$

Step 4.3.8

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

$4 x - \frac{x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [\frac{5}{x^{2}}] + \frac{d}{d x} [6]$

Step 4.3.9

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

$4 x - \frac{1}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [\frac{5}{x^{2}}] + \frac{d}{d x} [6]$

Step 4.4

Evaluate $\frac{d}{d x} [\frac{5}{x^{2}}]$ .

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Step 4.4.1

Since $5$ is constant with respect to $x$ , the derivative of $\frac{5}{x^{2}}$ with respect to $x$ is $5 \frac{d}{d x} [\frac{1}{x^{2}}]$ .

$4 x - \frac{1}{2 x^{\frac{1}{2}}} + 5 \frac{d}{d x} [\frac{1}{x^{2}}] + \frac{d}{d x} [6]$

Step 4.4.2

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

$4 x - \frac{1}{2 x^{\frac{1}{2}}} + 5 \frac{d}{d x} [{(x^{2})}^{- 1}] + \frac{d}{d x} [6]$

Step 4.4.3

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

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Step 4.4.3.1

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

$4 x - \frac{1}{2 x^{\frac{1}{2}}} + 5 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [6]$

Step 4.4.3.2

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

$4 x - \frac{1}{2 x^{\frac{1}{2}}} + 5 (- u^{- 2} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [6]$

Step 4.4.3.3

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

$4 x - \frac{1}{2 x^{\frac{1}{2}}} + 5 (- {(x^{2})}^{- 2} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [6]$

Step 4.4.4

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

$4 x - \frac{1}{2 x^{\frac{1}{2}}} + 5 (- {(x^{2})}^{- 2} (2 x)) + \frac{d}{d x} [6]$

Step 4.4.5

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

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Step 4.4.5.1

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

$4 x - \frac{1}{2 x^{\frac{1}{2}}} + 5 (- x^{2 \cdot - 2} (2 x)) + \frac{d}{d x} [6]$

Step 4.4.5.2

Multiply $2$ by $- 2$ .

$4 x - \frac{1}{2 x^{\frac{1}{2}}} + 5 (- x^{- 4} (2 x)) + \frac{d}{d x} [6]$

Step 4.4.6

Multiply $2$ by $- 1$ .

$4 x - \frac{1}{2 x^{\frac{1}{2}}} + 5 (- 2 x^{- 4} x) + \frac{d}{d x} [6]$

Step 4.4.7

Raise $x$ to the power of $1$ .

$4 x - \frac{1}{2 x^{\frac{1}{2}}} + 5 (- 2 (x^{1} x^{- 4})) + \frac{d}{d x} [6]$

Step 4.4.8

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

$4 x - \frac{1}{2 x^{\frac{1}{2}}} + 5 (- 2 x^{1 - 4}) + \frac{d}{d x} [6]$

Step 4.4.9

Subtract $4$ from $1$ .

$4 x - \frac{1}{2 x^{\frac{1}{2}}} + 5 (- 2 x^{- 3}) + \frac{d}{d x} [6]$

Step 4.4.10

Multiply $- 2$ by $5$ .

$4 x - \frac{1}{2 x^{\frac{1}{2}}} - 10 x^{- 3} + \frac{d}{d x} [6]$

Step 4.5

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$4 x - \frac{1}{2 x^{\frac{1}{2}}} - 10 x^{- 3} + 0$

Step 4.6

Simplify.

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Step 4.6.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$4 x - \frac{1}{2 x^{\frac{1}{2}}} - 10 \frac{1}{x^{3}} + 0$

Step 4.6.2

Combine terms.

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Step 4.6.2.1

Combine $- 10$ and $\frac{1}{x^{3}}$ .

$4 x - \frac{1}{2 x^{\frac{1}{2}}} + \frac{- 10}{x^{3}} + 0$

Step 4.6.2.2

Move the negative in front of the fraction.

$4 x - \frac{1}{2 x^{\frac{1}{2}}} - \frac{10}{x^{3}} + 0$

Step 4.6.2.3

Add $4 x - \frac{1}{2 x^{\frac{1}{2}}} - \frac{10}{x^{3}}$ and $0$ .

$4 x - \frac{1}{2 x^{\frac{1}{2}}} - \frac{10}{x^{3}}$

Step 4.6.3

Reorder terms.

$4 x - \frac{10}{x^{3}} - \frac{1}{2 x^{\frac{1}{2}}}$

Step 5

Reform the equation by setting the left side equal to the right side.

$y' = 4 x - \frac{10}{x^{3}} - \frac{1}{2 x^{\frac{1}{2}}}$

Step 6

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = 4 x - \frac{10}{x^{3}} - \frac{1}{2 x^{\frac{1}{2}}}$