Calculus Examples

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

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = 2 x^{2} + 1$ and $g (x) = \tan (x^{3})$ .

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Multiply $2$ by $2$ .

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

Step 2.5

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

$\frac{\tan (x^{3}) (4 x + 0) - (2 x^{2} + 1) \frac{d}{d x} [\tan (x^{3})]}{\tan^{2} (x^{3})}$

Step 2.6

Add $4 x$ and $0$ .

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

Step 3

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

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Step 3.1

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

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

Step 3.2

The derivative of $\tan (u)$ with respect to $u$ is $\sec^{2} (u)$ .

$\frac{\tan (x^{3}) (4 x) - (2 x^{2} + 1) (\sec^{2} (u) \frac{d}{d x} [x^{3}])}{\tan^{2} (x^{3})}$

Step 3.3

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

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

Step 4

Differentiate using the Power Rule.

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Step 4.1

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

$\frac{\tan (x^{3}) (4 x) - (2 x^{2} + 1) (\sec^{2} (x^{3}) (3 x^{2}))}{\tan^{2} (x^{3})}$

Step 4.2

Multiply $3$ by $- 1$ .

$\frac{\tan (x^{3}) (4 x) - 3 (2 x^{2} + 1) (\sec^{2} (x^{3}) (x^{2}))}{\tan^{2} (x^{3})}$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Apply the distributive property.

$\frac{\tan (x^{3}) (4 x) - 3 (2 x^{2}) (\sec^{2} (x^{3}) x^{2}) - 3 \cdot 1 (\sec^{2} (x^{3}) x^{2})}{\tan^{2} (x^{3})}$

Step 5.3

Simplify the numerator.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

Rewrite using the commutative property of multiplication.

$\frac{4 \tan (x^{3}) x - 3 (2 x^{2}) (\sec^{2} (x^{3}) x^{2}) - 3 \cdot 1 (\sec^{2} (x^{3}) x^{2})}{\tan^{2} (x^{3})}$

Step 5.3.1.2

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

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Step 5.3.1.2.1

Move $x^{2}$ .

$\frac{4 \tan (x^{3}) x - 3 (2 (x^{2} x^{2})) \sec^{2} (x^{3}) - 3 \cdot 1 (\sec^{2} (x^{3}) x^{2})}{\tan^{2} (x^{3})}$

Step 5.3.1.2.2

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

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

Step 5.3.1.2.3

Add $2$ and $2$ .

$\frac{4 \tan (x^{3}) x - 3 (2 x^{4}) \sec^{2} (x^{3}) - 3 \cdot 1 (\sec^{2} (x^{3}) x^{2})}{\tan^{2} (x^{3})}$

Step 5.3.1.3

Multiply $2$ by $- 3$ .

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

Step 5.3.1.4

Multiply $- 3$ by $1$ .

$\frac{4 \tan (x^{3}) x - 6 x^{4} \sec^{2} (x^{3}) - 3 \sec^{2} (x^{3}) x^{2}}{\tan^{2} (x^{3})}$

Step 5.3.2

Reorder factors in $4 \tan (x^{3}) x - 6 x^{4} \sec^{2} (x^{3}) - 3 \sec^{2} (x^{3}) x^{2}$ .

$\frac{4 x \tan (x^{3}) - 6 x^{4} \sec^{2} (x^{3}) - 3 x^{2} \sec^{2} (x^{3})}{\tan^{2} (x^{3})}$