Calculus Examples

$\frac{\sin (2 x^{3})}{4 x^{4} - 5}$

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 $f (x) = \sin (2 x^{3})$ and $g (x) = 4 x^{4} - 5$ .

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = 2 x^{3}$ .

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To apply the Chain Rule, set $u$ as $2 x^{3}$ .

$\frac{(4 x^{4} - 5) (\frac{d}{d u} [\sin (u)] \frac{d}{d x} [2 x^{3}]) - \sin (2 x^{3}) \frac{d}{d x} [4 x^{4} - 5]}{{(4 x^{4} - 5)}^{2}}$

The derivative of $\sin (u)$ with respect to $u$ is $\cos (u)$ .

$\frac{(4 x^{4} - 5) (\cos (u) \frac{d}{d x} [2 x^{3}]) - \sin (2 x^{3}) \frac{d}{d x} [4 x^{4} - 5]}{{(4 x^{4} - 5)}^{2}}$

Replace all occurrences of $u$ with $2 x^{3}$ .

$\frac{(4 x^{4} - 5) (\cos (2 x^{3}) \frac{d}{d x} [2 x^{3}]) - \sin (2 x^{3}) \frac{d}{d x} [4 x^{4} - 5]}{{(4 x^{4} - 5)}^{2}}$

Step 3

Differentiate.

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

$\frac{(4 x^{4} - 5) \cos (2 x^{3}) (2 \frac{d}{d x} [x^{3}]) - \sin (2 x^{3}) \frac{d}{d x} [4 x^{4} - 5]}{{(4 x^{4} - 5)}^{2}}$

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

$\frac{(4 x^{4} - 5) \cos (2 x^{3}) (2 (3 x^{2})) - \sin (2 x^{3}) \frac{d}{d x} [4 x^{4} - 5]}{{(4 x^{4} - 5)}^{2}}$

Multiply $3$ by $2$ .

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

By the Sum Rule, the derivative of $4 x^{4} - 5$ with respect to $x$ is $\frac{d}{d x} [4 x^{4}] + \frac{d}{d x} [- 5]$ .

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

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

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

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

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

Multiply $4$ by $4$ .

$\frac{(4 x^{4} - 5) \cos (2 x^{3}) (6 x^{2}) - \sin (2 x^{3}) (16 x^{3} + \frac{d}{d x} [- 5])}{{(4 x^{4} - 5)}^{2}}$

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

$\frac{(4 x^{4} - 5) \cos (2 x^{3}) (6 x^{2}) - \sin (2 x^{3}) (16 x^{3} + 0)}{{(4 x^{4} - 5)}^{2}}$

Simplify the expression.

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Add $16 x^{3}$ and $0$ .

$\frac{(4 x^{4} - 5) \cos (2 x^{3}) (6 x^{2}) - \sin (2 x^{3}) (16 x^{3})}{{(4 x^{4} - 5)}^{2}}$

Multiply $16$ by $- 1$ .

$\frac{(4 x^{4} - 5) \cos (2 x^{3}) (6 x^{2}) - 16 \sin (2 x^{3}) x^{3}}{{(4 x^{4} - 5)}^{2}}$

Step 4

Simplify.

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Apply the distributive property.

$\frac{(4 x^{4} \cos (2 x^{3}) - 5 \cos (2 x^{3})) (6 x^{2}) - 16 \sin (2 x^{3}) x^{3}}{{(4 x^{4} - 5)}^{2}}$

Apply the distributive property.

$\frac{4 x^{4} \cos (2 x^{3}) (6 x^{2}) - 5 \cos (2 x^{3}) (6 x^{2}) - 16 \sin (2 x^{3}) x^{3}}{{(4 x^{4} - 5)}^{2}}$

Simplify the numerator.

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Simplify each term.

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Multiply $x^{4}$ by $x^{2}$ by adding the exponents.

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Move $x^{2}$ .

$\frac{4 (x^{2} x^{4}) \cos (2 x^{3}) \cdot 6 - 5 \cos (2 x^{3}) (6 x^{2}) - 16 \sin (2 x^{3}) x^{3}}{{(4 x^{4} - 5)}^{2}}$

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

$\frac{4 x^{2 + 4} \cos (2 x^{3}) \cdot 6 - 5 \cos (2 x^{3}) (6 x^{2}) - 16 \sin (2 x^{3}) x^{3}}{{(4 x^{4} - 5)}^{2}}$

Add $2$ and $4$ .

$\frac{4 x^{6} \cos (2 x^{3}) \cdot 6 - 5 \cos (2 x^{3}) (6 x^{2}) - 16 \sin (2 x^{3}) x^{3}}{{(4 x^{4} - 5)}^{2}}$

Multiply $6$ by $4$ .

$\frac{24 x^{6} \cos (2 x^{3}) - 5 \cos (2 x^{3}) (6 x^{2}) - 16 \sin (2 x^{3}) x^{3}}{{(4 x^{4} - 5)}^{2}}$

Multiply $6$ by $- 5$ .

$\frac{24 x^{6} \cos (2 x^{3}) - 30 \cos (2 x^{3}) x^{2} - 16 \sin (2 x^{3}) x^{3}}{{(4 x^{4} - 5)}^{2}}$

Reorder factors in $24 x^{6} \cos (2 x^{3}) - 30 \cos (2 x^{3}) x^{2} - 16 \sin (2 x^{3}) x^{3}$ .

$\frac{24 x^{6} \cos (2 x^{3}) - 30 x^{2} \cos (2 x^{3}) - 16 x^{3} \sin (2 x^{3})}{{(4 x^{4} - 5)}^{2}}$