Calculus Examples

$f (x) = 2 x^{2} \arctan (5 x^{3})$

Step 1

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

$2 \frac{d}{d x} [x^{2} \arctan (5 x^{3})]$

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 $f (x) = x^{2}$ and $g (x) = \arctan (5 x^{3})$ .

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

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \arctan (x)$ and $g (x) = 5 x^{3}$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $5 x^{3}$ .

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

Step 3.2

The derivative of $\arctan (u)$ with respect to $u$ is $\frac{1}{1 + u^{2}}$ .

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

Step 3.3

Replace all occurrences of $u$ with $5 x^{3}$ .

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

Step 4

Differentiate.

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Step 4.1

Factor $5$ out of $5 x^{3}$ .

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

Step 4.2

Combine fractions.

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Step 4.2.1

Simplify the expression.

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Step 4.2.1.1

Apply the product rule to $5 (x^{3})$ .

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

Step 4.2.1.2

Raise $5$ to the power of $2$ .

$2 (x^{2} (\frac{1}{1 + 25 {(x^{3})}^{2}} \frac{d}{d x} [5 x^{3}]) + \arctan (5 x^{3}) \frac{d}{d x} [x^{2}])$

Step 4.2.1.3

Multiply the exponents in ${(x^{3})}^{2}$ .

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Step 4.2.1.3.1

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

$2 (x^{2} (\frac{1}{1 + 25 x^{3 \cdot 2}} \frac{d}{d x} [5 x^{3}]) + \arctan (5 x^{3}) \frac{d}{d x} [x^{2}])$

Step 4.2.1.3.2

Multiply $3$ by $2$ .

$2 (x^{2} (\frac{1}{1 + 25 x^{6}} \frac{d}{d x} [5 x^{3}]) + \arctan (5 x^{3}) \frac{d}{d x} [x^{2}])$

Step 4.2.2

Combine $\frac{1}{1 + 25 x^{6}}$ and $x^{2}$ .

$2 (\frac{x^{2}}{1 + 25 x^{6}} \frac{d}{d x} [5 x^{3}] + \arctan (5 x^{3}) \frac{d}{d x} [x^{2}])$

Step 4.3

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{3}$ with respect to $x$ is $5 \frac{d}{d x} [x^{3}]$ .

$2 (\frac{x^{2}}{1 + 25 x^{6}} (5 \frac{d}{d x} [x^{3}]) + \arctan (5 x^{3}) \frac{d}{d x} [x^{2}])$

Step 4.4

Combine $5$ and $\frac{x^{2}}{1 + 25 x^{6}}$ .

$2 (\frac{5 x^{2}}{1 + 25 x^{6}} \frac{d}{d x} [x^{3}] + \arctan (5 x^{3}) \frac{d}{d x} [x^{2}])$

Step 4.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$2 (\frac{5 x^{2}}{1 + 25 x^{6}} (3 x^{2}) + \arctan (5 x^{3}) \frac{d}{d x} [x^{2}])$

Step 4.6

Combine fractions.

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Step 4.6.1

Combine $3$ and $\frac{5 x^{2}}{1 + 25 x^{6}}$ .

$2 (\frac{3 (5 x^{2})}{1 + 25 x^{6}} x^{2} + \arctan (5 x^{3}) \frac{d}{d x} [x^{2}])$

Step 4.6.2

Multiply $5$ by $3$ .

$2 (\frac{15 x^{2}}{1 + 25 x^{6}} x^{2} + \arctan (5 x^{3}) \frac{d}{d x} [x^{2}])$

Step 4.6.3

Combine $\frac{15 x^{2}}{1 + 25 x^{6}}$ and $x^{2}$ .

$2 (\frac{15 x^{2} x^{2}}{1 + 25 x^{6}} + \arctan (5 x^{3}) \frac{d}{d x} [x^{2}])$

Step 5

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 5.1

Move $x^{2}$ .

$2 (\frac{15 (x^{2} x^{2})}{1 + 25 x^{6}} + \arctan (5 x^{3}) \frac{d}{d x} [x^{2}])$

Step 5.2

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

$2 (\frac{15 x^{2 + 2}}{1 + 25 x^{6}} + \arctan (5 x^{3}) \frac{d}{d x} [x^{2}])$

Step 5.3

Add $2$ and $2$ .

$2 (\frac{15 x^{4}}{1 + 25 x^{6}} + \arctan (5 x^{3}) \frac{d}{d x} [x^{2}])$

Step 6

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

$2 (\frac{15 x^{4}}{1 + 25 x^{6}} + \arctan (5 x^{3}) (2 x))$

Step 7

To write $\arctan (5 x^{3}) (2 x)$ as a fraction with a common denominator, multiply by $\frac{1 + 25 x^{6}}{1 + 25 x^{6}}$ .

$2 (\frac{15 x^{4}}{1 + 25 x^{6}} + \frac{\arctan (5 x^{3}) (2 x) (1 + 25 x^{6})}{1 + 25 x^{6}})$

Step 8

Combine the numerators over the common denominator.

$2 \frac{15 x^{4} + \arctan (5 x^{3}) (2 x) (1 + 25 x^{6})}{1 + 25 x^{6}}$

Step 9

Combine $2$ and $\frac{15 x^{4} + \arctan (5 x^{3}) (2 x) (1 + 25 x^{6})}{1 + 25 x^{6}}$ .

$\frac{2 (15 x^{4} + \arctan (5 x^{3}) (2 x) (1 + 25 x^{6}))}{1 + 25 x^{6}}$

Step 10

Simplify.

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Step 10.1

Apply the distributive property.

$\frac{2 (15 x^{4} + \arctan (5 x^{3}) (2 x) \cdot 1 + \arctan (5 x^{3}) (2 x) (25 x^{6}))}{1 + 25 x^{6}}$

Step 10.2

Apply the distributive property.

$\frac{2 (15 x^{4}) + 2 (\arctan (5 x^{3}) (2 x) \cdot 1) + 2 (\arctan (5 x^{3}) (2 x) (25 x^{6}))}{1 + 25 x^{6}}$

Step 10.3

Simplify the numerator.

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Step 10.3.1

Simplify each term.

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Step 10.3.1.1

Multiply $15$ by $2$ .

$\frac{30 x^{4} + 2 (\arctan (5 x^{3}) (2 x) \cdot 1) + 2 (\arctan (5 x^{3}) (2 x) (25 x^{6}))}{1 + 25 x^{6}}$

Step 10.3.1.2

Rewrite using the commutative property of multiplication.

$\frac{30 x^{4} + 2 (2 \arctan (5 x^{3}) x \cdot 1) + 2 (\arctan (5 x^{3}) (2 x) (25 x^{6}))}{1 + 25 x^{6}}$

Step 10.3.1.3

Multiply $2$ by $1$ .

$\frac{30 x^{4} + 2 (2 \arctan (5 x^{3}) x) + 2 (\arctan (5 x^{3}) (2 x) (25 x^{6}))}{1 + 25 x^{6}}$

Step 10.3.1.4

Multiply $2$ by $2$ .

$\frac{30 x^{4} + 4 (\arctan (5 x^{3}) x) + 2 (\arctan (5 x^{3}) (2 x) (25 x^{6}))}{1 + 25 x^{6}}$

Step 10.3.1.5

Multiply $x$ by $x^{6}$ by adding the exponents.

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Step 10.3.1.5.1

Move $x^{6}$ .

$\frac{30 x^{4} + 4 \arctan (5 x^{3}) x + 2 (\arctan (5 x^{3}) (2 (x^{6} x)) \cdot 25)}{1 + 25 x^{6}}$

Step 10.3.1.5.2

Multiply $x^{6}$ by $x$ .

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Step 10.3.1.5.2.1

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

$\frac{30 x^{4} + 4 \arctan (5 x^{3}) x + 2 (\arctan (5 x^{3}) (2 (x^{6} x^{1})) \cdot 25)}{1 + 25 x^{6}}$

Step 10.3.1.5.2.2

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

$\frac{30 x^{4} + 4 \arctan (5 x^{3}) x + 2 (\arctan (5 x^{3}) (2 x^{6 + 1}) \cdot 25)}{1 + 25 x^{6}}$

Step 10.3.1.5.3

Add $6$ and $1$ .

$\frac{30 x^{4} + 4 \arctan (5 x^{3}) x + 2 (\arctan (5 x^{3}) (2 x^{7}) \cdot 25)}{1 + 25 x^{6}}$

Step 10.3.1.6

Rewrite using the commutative property of multiplication.

$\frac{30 x^{4} + 4 \arctan (5 x^{3}) x + 2 (2 \arctan (5 x^{3}) x^{7} \cdot 25)}{1 + 25 x^{6}}$

Step 10.3.1.7

Multiply $25$ by $2$ .

$\frac{30 x^{4} + 4 \arctan (5 x^{3}) x + 2 (50 \arctan (5 x^{3}) x^{7})}{1 + 25 x^{6}}$

Step 10.3.1.8

Multiply $50$ by $2$ .

$\frac{30 x^{4} + 4 \arctan (5 x^{3}) x + 100 \arctan (5 x^{3}) x^{7}}{1 + 25 x^{6}}$

Step 10.3.2

Reorder factors in $30 x^{4} + 4 \arctan (5 x^{3}) x + 100 \arctan (5 x^{3}) x^{7}$ .

$\frac{30 x^{4} + 4 x \arctan (5 x^{3}) + 100 x^{7} \arctan (5 x^{3})}{1 + 25 x^{6}}$

Step 10.4

Reorder terms.

$\frac{30 x^{4} + 4 x \arctan (5 x^{3}) + 100 x^{7} \arctan (5 x^{3})}{25 x^{6} + 1}$