Calculus Examples

$y = 2 x (\sqrt{x} - x^{2} + 3 x - 5)$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

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

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

Step 1.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x (x^{\frac{1}{2}} - x^{2} + 3 x - 5)$ with respect to $x$ is $2 \frac{d}{d x} [x (x^{\frac{1}{2}} - x^{2} + 3 x - 5)]$ .

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

Step 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 $f (x) = x$ and $g (x) = x^{\frac{1}{2}} - x^{2} + 3 x - 5$ .

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

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $x^{\frac{1}{2}} - x^{2} + 3 x - 5$ with respect to $x$ is $\frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 5]$ .

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

Step 3.2

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

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

Step 4

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

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

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

Step 7.2

Subtract $2$ from $1$ .

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

Step 8

Combine fractions.

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Step 8.1

Move the negative in front of the fraction.

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

Step 8.2

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

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

Step 8.3

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

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

Step 9

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

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

Step 10

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

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

Step 11

Multiply $2$ by $- 1$ .

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

Step 12

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

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

Step 13

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

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

Step 14

Multiply $3$ by $1$ .

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

Step 15

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$2 (x (\frac{1}{2 x^{\frac{1}{2}}} - 2 x + 3 + 0) + (x^{\frac{1}{2}} - x^{2} + 3 x - 5) \frac{d}{d x} [x])$

Step 16

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

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

Step 17

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

$2 (x (\frac{1}{2 x^{\frac{1}{2}}} - 2 x + 3) + (x^{\frac{1}{2}} - x^{2} + 3 x - 5) \cdot 1)$

Step 18

Multiply $x^{\frac{1}{2}} - x^{2} + 3 x - 5$ by $1$ .

$2 (x (\frac{1}{2 x^{\frac{1}{2}}} - 2 x + 3) + x^{\frac{1}{2}} - x^{2} + 3 x - 5)$

Step 19

Simplify.

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Step 19.1

Apply the distributive property.

$2 (x \frac{1}{2 x^{\frac{1}{2}}} + x (- 2 x) + x \cdot 3 + x^{\frac{1}{2}} - x^{2} + 3 x - 5)$

Step 19.2

Apply the distributive property.

$2 (x \frac{1}{2 x^{\frac{1}{2}}}) + 2 (x (- 2 x)) + 2 (x \cdot 3) + 2 x^{\frac{1}{2}} + 2 (- x^{2}) + 2 (3 x) + 2 \cdot - 5$

Step 19.3

Combine terms.

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Step 19.3.1

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

$2 \frac{x}{2 x^{\frac{1}{2}}} + 2 (x (- 2 x)) + 2 (x \cdot 3) + 2 x^{\frac{1}{2}} + 2 (- x^{2}) + 2 (3 x) + 2 \cdot - 5$

Step 19.3.2

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

$2 \frac{x \cdot x^{- \frac{1}{2}}}{2} + 2 (x (- 2 x)) + 2 (x \cdot 3) + 2 x^{\frac{1}{2}} + 2 (- x^{2}) + 2 (3 x) + 2 \cdot - 5$

Step 19.3.3

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

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Step 19.3.3.1

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

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Step 19.3.3.1.1

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

$2 \frac{x^{1} x^{- \frac{1}{2}}}{2} + 2 (x (- 2 x)) + 2 (x \cdot 3) + 2 x^{\frac{1}{2}} + 2 (- x^{2}) + 2 (3 x) + 2 \cdot - 5$

Step 19.3.3.1.2

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

$2 \frac{x^{1 - \frac{1}{2}}}{2} + 2 (x (- 2 x)) + 2 (x \cdot 3) + 2 x^{\frac{1}{2}} + 2 (- x^{2}) + 2 (3 x) + 2 \cdot - 5$

Step 19.3.3.2

Write $1$ as a fraction with a common denominator.

$2 \frac{x^{\frac{2}{2} - \frac{1}{2}}}{2} + 2 (x (- 2 x)) + 2 (x \cdot 3) + 2 x^{\frac{1}{2}} + 2 (- x^{2}) + 2 (3 x) + 2 \cdot - 5$

Step 19.3.3.3

Combine the numerators over the common denominator.

$2 \frac{x^{\frac{2 - 1}{2}}}{2} + 2 (x (- 2 x)) + 2 (x \cdot 3) + 2 x^{\frac{1}{2}} + 2 (- x^{2}) + 2 (3 x) + 2 \cdot - 5$

Step 19.3.3.4

Subtract $1$ from $2$ .

$2 \frac{x^{\frac{1}{2}}}{2} + 2 (x (- 2 x)) + 2 (x \cdot 3) + 2 x^{\frac{1}{2}} + 2 (- x^{2}) + 2 (3 x) + 2 \cdot - 5$

Step 19.3.4

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

$\frac{2 x^{\frac{1}{2}}}{2} + 2 (x (- 2 x)) + 2 (x \cdot 3) + 2 x^{\frac{1}{2}} + 2 (- x^{2}) + 2 (3 x) + 2 \cdot - 5$

Step 19.3.5

Cancel the common factor.

$\frac{2 x^{\frac{1}{2}}}{2} + 2 (x (- 2 x)) + 2 (x \cdot 3) + 2 x^{\frac{1}{2}} + 2 (- x^{2}) + 2 (3 x) + 2 \cdot - 5$

Step 19.3.6

Divide $x^{\frac{1}{2}}$ by $1$ .

$x^{\frac{1}{2}} + 2 (x (- 2 x)) + 2 (x \cdot 3) + 2 x^{\frac{1}{2}} + 2 (- x^{2}) + 2 (3 x) + 2 \cdot - 5$

Step 19.3.7

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

$x^{\frac{1}{2}} + 2 (- 2 (x^{1} x)) + 2 (x \cdot 3) + 2 x^{\frac{1}{2}} + 2 (- x^{2}) + 2 (3 x) + 2 \cdot - 5$

Step 19.3.8

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

$x^{\frac{1}{2}} + 2 (- 2 (x^{1} x^{1})) + 2 (x \cdot 3) + 2 x^{\frac{1}{2}} + 2 (- x^{2}) + 2 (3 x) + 2 \cdot - 5$

Step 19.3.9

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

$x^{\frac{1}{2}} + 2 (- 2 x^{1 + 1}) + 2 (x \cdot 3) + 2 x^{\frac{1}{2}} + 2 (- x^{2}) + 2 (3 x) + 2 \cdot - 5$

Step 19.3.10

Add $1$ and $1$ .

$x^{\frac{1}{2}} + 2 (- 2 x^{2}) + 2 (x \cdot 3) + 2 x^{\frac{1}{2}} + 2 (- x^{2}) + 2 (3 x) + 2 \cdot - 5$

Step 19.3.11

Multiply $- 2$ by $2$ .

$x^{\frac{1}{2}} - 4 x^{2} + 2 (x \cdot 3) + 2 x^{\frac{1}{2}} + 2 (- x^{2}) + 2 (3 x) + 2 \cdot - 5$

Step 19.3.12

Move $3$ to the left of $x$ .

$x^{\frac{1}{2}} - 4 x^{2} + 2 (3 \cdot x) + 2 x^{\frac{1}{2}} + 2 (- x^{2}) + 2 (3 x) + 2 \cdot - 5$

Step 19.3.13

Multiply $3$ by $2$ .

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

Step 19.3.14

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

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

Step 19.3.15

Multiply $- 1$ by $2$ .

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

Step 19.3.16

Subtract $2 x^{2}$ from $- 4 x^{2}$ .

$- 6 x^{2} + 6 x + 3 x^{\frac{1}{2}} + 2 (3 x) + 2 \cdot - 5$

Step 19.3.17

Multiply $3$ by $2$ .

$- 6 x^{2} + 6 x + 3 x^{\frac{1}{2}} + 6 x + 2 \cdot - 5$

Step 19.3.18

Add $6 x$ and $6 x$ .

$- 6 x^{2} + 12 x + 3 x^{\frac{1}{2}} + 2 \cdot - 5$

Step 19.3.19

Multiply $2$ by $- 5$ .

$- 6 x^{2} + 12 x + 3 x^{\frac{1}{2}} - 10$