Calculus Examples

$\frac{3 x^{7} - x^{3}}{x}$

Step 1

Simplify terms.

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Step 1.1

Factor $x^{3}$ out of $3 x^{7} - x^{3}$ .

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Step 1.1.1

Factor $x^{3}$ out of $3 x^{7}$ .

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

Step 1.1.2

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

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

Step 1.1.3

Factor $x^{3}$ out of $x^{3} (3 x^{4}) + x^{3} \cdot - 1$ .

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

Step 1.2

Cancel the common factor of $x^{3}$ and $x$ .

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Step 1.2.1

Factor $x$ out of $x^{3} (3 x^{4} - 1)$ .

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

Step 1.2.2

Cancel the common factors.

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Step 1.2.2.1

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

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

Step 1.2.2.2

Factor $x$ out of $x^{1}$ .

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

Step 1.2.2.3

Cancel the common factor.

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

Step 1.2.2.4

Rewrite the expression.

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

Step 1.2.2.5

Divide $x^{2} (3 x^{4} - 1)$ by $1$ .

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

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 $g (x) = 3 x^{4} - 1$ .

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

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $3 x^{4} - 1$ with respect to $x$ is $\frac{d}{d x} [3 x^{4}] + \frac{d}{d x} [- 1]$ .

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

Step 3.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{4}$ with respect to $x$ is $3 \frac{d}{d x} [x^{4}]$ .

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

Step 3.3

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

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

Step 3.4

Multiply $4$ by $3$ .

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

Step 3.5

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

$x^{2} (12 x^{3} + 0) + (3 x^{4} - 1) \frac{d}{d x} [x^{2}]$

Step 3.6

Add $12 x^{3}$ and $0$ .

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

Step 4

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

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Step 4.1

Move $x^{3}$ .

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

Step 4.2

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

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

Step 4.3

Add $3$ and $2$ .

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

Step 5

Move $12$ to the left of $x^{5}$ .

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

Step 6

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

$12 x^{5} + (3 x^{4} - 1) (2 x)$

Step 7

Move $2$ to the left of $3 x^{4} - 1$ .

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

Step 8

Simplify.

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Step 8.1

Apply the distributive property.

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

Step 8.2

Apply the distributive property.

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

Step 8.3

Combine terms.

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Step 8.3.1

Multiply $3$ by $2$ .

$12 x^{5} + 6 x^{4} x + 2 \cdot - 1 x$

Step 8.3.2

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

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

Step 8.3.3

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

$12 x^{5} + 6 x^{1 + 4} + 2 \cdot - 1 x$

Step 8.3.4

Add $1$ and $4$ .

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

Step 8.3.5

Multiply $2$ by $- 1$ .

$12 x^{5} + 6 x^{5} - 2 x$

Step 8.3.6

Add $12 x^{5}$ and $6 x^{5}$ .

$18 x^{5} - 2 x$