Calculus Examples

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

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 $g (x) = - 3 x^{2} - 2$ .

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

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $- 3 x^{2} - 2$ with respect to $x$ is $\frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [- 2]$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Multiply $2$ by $- 3$ .

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

Step 2.5

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

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

Step 2.6

Add $- 6 x$ and $0$ .

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

Step 3

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

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Step 3.1

Move $x$ .

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

Step 3.2

Multiply $x$ by $x^{2}$ .

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.3

Add $1$ and $2$ .

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

Step 4

Move $- 6$ to the left of $x^{3}$ .

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

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

Step 7.2

Apply the distributive property.

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

Step 7.3

Combine terms.

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Step 7.3.1

Multiply $- 3$ by $2$ .

$- 6 x^{3} - 6 x^{2} x + 2 \cdot - 2 x$

Step 7.3.2

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

$- 6 x^{3} - 6 (x^{1} x^{2}) + 2 \cdot - 2 x$

Step 7.3.3

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

$- 6 x^{3} - 6 x^{1 + 2} + 2 \cdot - 2 x$

Step 7.3.4

Add $1$ and $2$ .

$- 6 x^{3} - 6 x^{3} + 2 \cdot - 2 x$

Step 7.3.5

Multiply $2$ by $- 2$ .

$- 6 x^{3} - 6 x^{3} - 4 x$

Step 7.3.6

Subtract $6 x^{3}$ from $- 6 x^{3}$ .

$- 12 x^{3} - 4 x$