Calculus Examples

$f (x) = x^{3} (5 - 3 x^{2})$

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 $g (x) = 5 - 3 x^{2}$ .

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

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $5 - 3 x^{2}$ with respect to $x$ is $\frac{d}{d x} [5] + \frac{d}{d x} [- 3 x^{2}]$ .

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

Step 2.2

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$x^{3} (0 + \frac{d}{d x} [- 3 x^{2}]) + (5 - 3 x^{2}) \frac{d}{d x} [x^{3}]$

Step 2.3

Add $0$ and $\frac{d}{d x} [- 3 x^{2}]$ .

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Multiply $2$ by $- 3$ .

$x^{3} (- 6 x) + (5 - 3 x^{2}) \frac{d}{d x} [x^{3}]$

Step 3

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

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Step 3.1

Move $x$ .

$x \cdot x^{3} \cdot - 6 + (5 - 3 x^{2}) \frac{d}{d x} [x^{3}]$

Step 3.2

Multiply $x$ by $x^{3}$ .

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.3

Add $1$ and $3$ .

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

Step 4

Move $- 6$ to the left of $x^{4}$ .

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

Step 5

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

$- 6 x^{4} + (5 - 3 x^{2}) (3 x^{2})$

Step 6

Move $3$ to the left of $5 - 3 x^{2}$ .

$- 6 x^{4} + 3 \cdot (5 - 3 x^{2}) x^{2}$

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

$- 6 x^{4} + (3 \cdot 5 + 3 (- 3 x^{2})) x^{2}$

Step 7.2

Apply the distributive property.

$- 6 x^{4} + 3 \cdot 5 x^{2} + 3 (- 3 x^{2}) x^{2}$

Step 7.3

Combine terms.

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Step 7.3.1

Multiply $3$ by $5$ .

$- 6 x^{4} + 15 x^{2} + 3 (- 3 x^{2}) x^{2}$

Step 7.3.2

Multiply $- 3$ by $3$ .

$- 6 x^{4} + 15 x^{2} - 9 x^{2} x^{2}$

Step 7.3.3

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

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Step 7.3.3.1

Move $x^{2}$ .

$- 6 x^{4} + 15 x^{2} - 9 (x^{2} x^{2})$

Step 7.3.3.2

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

$- 6 x^{4} + 15 x^{2} - 9 x^{2 + 2}$

Step 7.3.3.3

Add $2$ and $2$ .

$- 6 x^{4} + 15 x^{2} - 9 x^{4}$

Step 7.3.4

Subtract $9 x^{4}$ from $- 6 x^{4}$ .

$- 15 x^{4} + 15 x^{2}$