Calculus Examples

$x^{2} (3 x^{3} - 1)$

Step 1

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

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

Step 2

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

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

Step 3

Evaluate $\frac{d}{d x} [3 x^{3}]$ .

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Multiply $3$ by $3$ .

$x^{2} (9 x^{2} + \frac{d}{d x} [- 1]) + (3 x^{3} - 1) \frac{d}{d x} [x^{2}]$

Step 4

Differentiate.

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Step 4.1

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

$x^{2} (9 x^{2} + 0) + (3 x^{3} - 1) \frac{d}{d x} [x^{2}]$

Step 4.2

Add $9 x^{2}$ and $0$ .

$x^{2} (9 x^{2}) + (3 x^{3} - 1) \frac{d}{d x} [x^{2}]$

Step 4.3

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

$x^{2} (9 x^{2}) + (3 x^{3} - 1) (2 x)$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$x^{2} (9 x^{2}) + 3 x^{3} (2 x) - 1 (2 x)$

Step 5.2

Combine terms.

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Step 5.2.1

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

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Step 5.2.1.1

Move $x^{2}$ .

$x^{2} x^{2} \cdot 9 + 3 x^{3} (2 x) - 1 (2 x)$

Step 5.2.1.2

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

$x^{2 + 2} \cdot 9 + 3 x^{3} (2 x) - 1 (2 x)$

Step 5.2.1.3

Add $2$ and $2$ .

$x^{4} \cdot 9 + 3 x^{3} (2 x) - 1 (2 x)$

Step 5.2.2

Move $9$ to the left of $x^{4}$ .

$9 \cdot x^{4} + 3 x^{3} (2 x) - 1 (2 x)$

Step 5.2.3

Multiply $2$ by $3$ .

$9 x^{4} + 6 x^{3} x - 1 (2 x)$

Step 5.2.4

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

$9 x^{4} + 6 (x^{1} x^{3}) - 1 (2 x)$

Step 5.2.5

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

$9 x^{4} + 6 x^{1 + 3} - 1 (2 x)$

Step 5.2.6

Add $1$ and $3$ .

$9 x^{4} + 6 x^{4} - 1 (2 x)$

Step 5.2.7

Multiply $2$ by $- 1$ .

$9 x^{4} + 6 x^{4} - 2 x$

Step 5.2.8

Add $9 x^{4}$ and $6 x^{4}$ .

$15 x^{4} - 2 x$