Calculus Examples

$f (x) = 3 x^{4} (2 x^{2} - 1)$

Step 1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{4} (2 x^{2} - 1)$ with respect to $x$ is $3 \frac{d}{d x} [x^{4} (2 x^{2} - 1)]$ .

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

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 $f (x) = x^{4}$ and $g (x) = 2 x^{2} - 1$ .

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

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $2 x^{2} - 1$ with respect to $x$ is $\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 1]$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Multiply $2$ by $2$ .

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

Step 3.5

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

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

Step 3.6

Add $4 x$ and $0$ .

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

Step 4

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

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Step 4.1

Move $x$ .

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

Step 4.2

Multiply $x$ by $x^{4}$ .

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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.3

Add $1$ and $4$ .

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

Step 5

Move $4$ to the left of $x^{5}$ .

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

Step 6

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

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

Step 7

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

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

Step 8

Simplify.

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Step 8.1

Apply the distributive property.

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

Step 8.2

Apply the distributive property.

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

Step 8.3

Apply the distributive property.

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

Step 8.4

Combine terms.

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Step 8.4.1

Multiply $4$ by $3$ .

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

Step 8.4.2

Multiply $2$ by $4$ .

$12 x^{5} + 3 (8 x^{2} x^{3}) + 3 (4 \cdot - 1 x^{3})$

Step 8.4.3

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

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Step 8.4.3.1

Move $x^{3}$ .

$12 x^{5} + 3 (8 (x^{3} x^{2})) + 3 (4 \cdot - 1 x^{3})$

Step 8.4.3.2

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

$12 x^{5} + 3 (8 x^{3 + 2}) + 3 (4 \cdot - 1 x^{3})$

Step 8.4.3.3

Add $3$ and $2$ .

$12 x^{5} + 3 (8 x^{5}) + 3 (4 \cdot - 1 x^{3})$

Step 8.4.4

Multiply $8$ by $3$ .

$12 x^{5} + 24 x^{5} + 3 (4 \cdot - 1 x^{3})$

Step 8.4.5

Multiply $4$ by $- 1$ .

$12 x^{5} + 24 x^{5} + 3 (- 4 x^{3})$

Step 8.4.6

Multiply $- 4$ by $3$ .

$12 x^{5} + 24 x^{5} - 12 x^{3}$

Step 8.4.7

Add $12 x^{5}$ and $24 x^{5}$ .

$36 x^{5} - 12 x^{3}$