Calculus Examples

$y = - x^{3} (3 x^{4} - 2)$

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

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

Step 2

By the Sum Rule, the derivative of $3 x^{4} - 2$ with respect to $x$ is $\frac{d}{d x} [3 x^{4}] + \frac{d}{d x} [- 2]$ .

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

Step 3

Evaluate $\frac{d}{d x} [3 x^{4}]$ .

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Multiply $4$ by $3$ .

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

Step 4

Differentiate.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

Multiply $3$ by $- 1$ .

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

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Combine terms.

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Step 5.2.1

Multiply $12$ by $- 1$ .

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

Step 5.2.2

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

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Step 5.2.2.1

Move $x^{3}$ .

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

Step 5.2.2.2

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

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

Step 5.2.2.3

Add $3$ and $3$ .

$- 12 x^{6} + 3 x^{4} (- 3 x^{2}) - 2 (- 3 x^{2})$

Step 5.2.3

Multiply $- 3$ by $3$ .

$- 12 x^{6} - 9 x^{4} x^{2} - 2 (- 3 x^{2})$

Step 5.2.4

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

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Step 5.2.4.1

Move $x^{2}$ .

$- 12 x^{6} - 9 (x^{2} x^{4}) - 2 (- 3 x^{2})$

Step 5.2.4.2

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

$- 12 x^{6} - 9 x^{2 + 4} - 2 (- 3 x^{2})$

Step 5.2.4.3

Add $2$ and $4$ .

$- 12 x^{6} - 9 x^{6} - 2 (- 3 x^{2})$

Step 5.2.5

Multiply $- 3$ by $- 2$ .

$- 12 x^{6} - 9 x^{6} + 6 x^{2}$

Step 5.2.6

Subtract $9 x^{6}$ from $- 12 x^{6}$ .

$- 21 x^{6} + 6 x^{2}$