Calculus Examples

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

Step 1

Factor $4$ out of $4 x$ .

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

Step 2

Simplify the expression.

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Step 2.1

Apply the product rule to $4 (x)$ .

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

Step 2.2

Raise $4$ to the power of $3$ .

$\frac{d}{d x} [64 x^{3} \cdot {(2 x)}^{6}]$

Step 3

Factor $2$ out of $2 x$ .

$\frac{d}{d x} [64 x^{3} \cdot {(2 (x))}^{6}]$

Step 4

Simplify the expression.

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Step 4.1

Apply the product rule to $2 (x)$ .

$\frac{d}{d x} [64 x^{3} \cdot (2^{6} x^{6})]$

Step 4.2

Raise $2$ to the power of $6$ .

$\frac{d}{d x} [64 x^{3} \cdot (64 x^{6})]$

Step 4.3

Multiply $64$ by $64$ .

$\frac{d}{d x} [4096 x^{3} \cdot x^{6}]$

Step 5

Multiply $x^{3}$ by $x^{6}$ by adding the exponents.

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Step 5.1

Move $x^{6}$ .

$\frac{d}{d x} [4096 (x^{6} x^{3})]$

Step 5.2

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

$\frac{d}{d x} [4096 x^{6 + 3}]$

Step 5.3

Add $6$ and $3$ .

$\frac{d}{d x} [4096 x^{9}]$

Step 6

Since $4096$ is constant with respect to $x$ , the derivative of $4096 x^{9}$ with respect to $x$ is $4096 \frac{d}{d x} [x^{9}]$ .

$4096 \frac{d}{d x} [x^{9}]$

Step 7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 9$ .

$4096 (9 x^{8})$

Step 8

Multiply $9$ by $4096$ .

$36864 x^{8}$