Calculus Examples

$\frac{{(x^{3} - 7)}^{4}}{2 x^{3}}$

Step 1

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{{(x^{3} - 7)}^{4}}{2 x^{3}}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [\frac{{(x^{3} - 7)}^{4}}{x^{3}}]$ .

$\frac{1}{2} \frac{d}{d x} [\frac{{(x^{3} - 7)}^{4}}{x^{3}}]$

Step 2

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 $f (x) = {(x^{3} - 7)}^{4}$ and $g (x) = x^{3}$ .

$\frac{1}{2} \cdot \frac{x^{3} \frac{d}{d x} [{(x^{3} - 7)}^{4}] - {(x^{3} - 7)}^{4} \frac{d}{d x} [x^{3}]}{{(x^{3})}^{2}}$

Step 3

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

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Step 3.1

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

$\frac{1}{2} \cdot \frac{x^{3} \frac{d}{d x} [{(x^{3} - 7)}^{4}] - {(x^{3} - 7)}^{4} \frac{d}{d x} [x^{3}]}{x^{3 \cdot 2}}$

Step 3.2

Multiply $3$ by $2$ .

$\frac{1}{2} \cdot \frac{x^{3} \frac{d}{d x} [{(x^{3} - 7)}^{4}] - {(x^{3} - 7)}^{4} \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{4}$ and $g (x) = x^{3} - 7$ .

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Step 4.1

To apply the Chain Rule, set $u$ as $x^{3} - 7$ .

$\frac{1}{2} \cdot \frac{x^{3} (\frac{d}{d u} [u^{4}] \frac{d}{d x} [x^{3} - 7]) - {(x^{3} - 7)}^{4} \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 4.2

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

$\frac{1}{2} \cdot \frac{x^{3} (4 u^{3} \frac{d}{d x} [x^{3} - 7]) - {(x^{3} - 7)}^{4} \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 4.3

Replace all occurrences of $u$ with $x^{3} - 7$ .

$\frac{1}{2} \cdot \frac{x^{3} (4 {(x^{3} - 7)}^{3} \frac{d}{d x} [x^{3} - 7]) - {(x^{3} - 7)}^{4} \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 5

Differentiate.

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Step 5.1

By the Sum Rule, the derivative of $x^{3} - 7$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 7]$ .

$\frac{1}{2} \cdot \frac{x^{3} (4 {(x^{3} - 7)}^{3} (\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 7])) - {(x^{3} - 7)}^{4} \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 5.2

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

$\frac{1}{2} \cdot \frac{x^{3} (4 {(x^{3} - 7)}^{3} (3 x^{2} + \frac{d}{d x} [- 7])) - {(x^{3} - 7)}^{4} \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 5.3

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7$ with respect to $x$ is $0$ .

$\frac{1}{2} \cdot \frac{x^{3} (4 {(x^{3} - 7)}^{3} (3 x^{2} + 0)) - {(x^{3} - 7)}^{4} \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 5.4

Simplify the expression.

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Step 5.4.1

Add $3 x^{2}$ and $0$ .

$\frac{1}{2} \cdot \frac{x^{3} (4 {(x^{3} - 7)}^{3} (3 x^{2})) - {(x^{3} - 7)}^{4} \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 5.4.2

Multiply $3$ by $4$ .

$\frac{1}{2} \cdot \frac{x^{3} (12 {(x^{3} - 7)}^{3} x^{2}) - {(x^{3} - 7)}^{4} \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 6

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

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Step 6.1

Move $x^{2}$ .

$\frac{1}{2} \cdot \frac{x^{2} x^{3} (12 {(x^{3} - 7)}^{3}) - {(x^{3} - 7)}^{4} \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 6.2

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

$\frac{1}{2} \cdot \frac{x^{2 + 3} (12 {(x^{3} - 7)}^{3}) - {(x^{3} - 7)}^{4} \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 6.3

Add $2$ and $3$ .

$\frac{1}{2} \cdot \frac{x^{5} (12 {(x^{3} - 7)}^{3}) - {(x^{3} - 7)}^{4} \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 7

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

$\frac{1}{2} \cdot \frac{x^{5} (12 {(x^{3} - 7)}^{3}) - {(x^{3} - 7)}^{4} (3 x^{2})}{x^{6}}$

Step 8

Combine fractions.

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Step 8.1

Multiply $3$ by $- 1$ .

$\frac{1}{2} \cdot \frac{x^{5} (12 {(x^{3} - 7)}^{3}) - 3 {(x^{3} - 7)}^{4} x^{2}}{x^{6}}$

Step 8.2

Multiply $\frac{1}{2}$ by $\frac{x^{5} (12 {(x^{3} - 7)}^{3}) - 3 {(x^{3} - 7)}^{4} x^{2}}{x^{6}}$ .

$\frac{x^{5} (12 {(x^{3} - 7)}^{3}) - 3 {(x^{3} - 7)}^{4} x^{2}}{2 x^{6}}$

Step 9

Simplify.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Factor $x^{2} \cdot 3 {(x^{3} - 7)}^{3}$ out of $x^{5} (12 {(x^{3} - 7)}^{3}) - 3 {(x^{3} - 7)}^{4} x^{2}$ .

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Step 9.1.1.1

Factor $x^{2} \cdot 3 {(x^{3} - 7)}^{3}$ out of $x^{5} (12 {(x^{3} - 7)}^{3})$ .

$\frac{x^{2} \cdot 3 {(x^{3} - 7)}^{3} (x^{3} \cdot (4)) - 3 {(x^{3} - 7)}^{4} x^{2}}{2 x^{6}}$

Step 9.1.1.2

Factor $x^{2} \cdot 3 {(x^{3} - 7)}^{3}$ out of $- 3 {(x^{3} - 7)}^{4} x^{2}$ .

$\frac{x^{2} \cdot 3 {(x^{3} - 7)}^{3} (x^{3} \cdot (4)) + x^{2} \cdot 3 {(x^{3} - 7)}^{3} (- (x^{3} - 7))}{2 x^{6}}$

Step 9.1.1.3

Factor $x^{2} \cdot 3 {(x^{3} - 7)}^{3}$ out of $x^{2} \cdot 3 {(x^{3} - 7)}^{3} (x^{3} \cdot (4)) + x^{2} \cdot 3 {(x^{3} - 7)}^{3} (- (x^{3} - 7))$ .

$\frac{x^{2} \cdot 3 {(x^{3} - 7)}^{3} (x^{3} \cdot (4) - (x^{3} - 7))}{2 x^{6}}$

Step 9.1.2

Move $4$ to the left of $x^{3}$ .

$\frac{x^{2} \cdot 3 {(x^{3} - 7)}^{3} (4 \cdot x^{3} - (x^{3} - 7))}{2 x^{6}}$

Step 9.1.3

Apply the distributive property.

$\frac{x^{2} \cdot 3 {(x^{3} - 7)}^{3} (4 x^{3} - x^{3} - - 7)}{2 x^{6}}$

Step 9.1.4

Multiply $- 1$ by $- 7$ .

$\frac{x^{2} \cdot 3 {(x^{3} - 7)}^{3} (4 x^{3} - x^{3} + 7)}{2 x^{6}}$

Step 9.1.5

Subtract $x^{3}$ from $4 x^{3}$ .

$\frac{x^{2} \cdot 3 {(x^{3} - 7)}^{3} (3 x^{3} + 7)}{2 x^{6}}$

Step 9.2

Combine terms.

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Step 9.2.1

Move $3$ to the left of $x^{2}$ .

$\frac{3 \cdot x^{2} {(x^{3} - 7)}^{3} (3 x^{3} + 7)}{2 x^{6}}$

Step 9.2.2

Cancel the common factor of $x^{2}$ and $x^{6}$ .

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Step 9.2.2.1

Factor $x^{2}$ out of $3 x^{2} {(x^{3} - 7)}^{3} (3 x^{3} + 7)$ .

$\frac{x^{2} (3 {(x^{3} - 7)}^{3} (3 x^{3} + 7))}{2 x^{6}}$

Step 9.2.2.2

Cancel the common factors.

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Step 9.2.2.2.1

Factor $x^{2}$ out of $2 x^{6}$ .

$\frac{x^{2} (3 {(x^{3} - 7)}^{3} (3 x^{3} + 7))}{x^{2} (2 x^{4})}$

Step 9.2.2.2.2

Cancel the common factor.

$\frac{x^{2} (3 {(x^{3} - 7)}^{3} (3 x^{3} + 7))}{x^{2} (2 x^{4})}$

Step 9.2.2.2.3

Rewrite the expression.

$\frac{3 {(x^{3} - 7)}^{3} (3 x^{3} + 7)}{2 x^{4}}$