Calculus Examples

(3 x^{3} + 1) {(4 x^{3} \cdot (2 x))}^{6}

$(3 x^{3} + 1) {(4 x^{3} \cdot (2 x))}^{6}$

Step 1

This derivative could not be completed using the chain rule. Mathway will use another method.

Step 2

Rewrite using the commutative property of multiplication.

\frac{d}{d X} [(3 x^{3} + 1) {(4 \cdot 2 x^{3} x)}^{6}]

$\frac{d}{d X} [(3 x^{3} + 1) {(4 \cdot 2 x^{3} x)}^{6}]$

Step 3

Multiply

x^{3}

$x^{3}$ by

x

$x$ by adding the exponents.

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Step 3.1

Move

x

$x$ .

\frac{d}{d X} [(3 x^{3} + 1) {(4 \cdot 2 (x \cdot x^{3}))}^{6}]

$\frac{d}{d X} [(3 x^{3} + 1) {(4 \cdot 2 (x \cdot x^{3}))}^{6}]$

Step 3.2

Multiply

x

$x$ by

x^{3}

$x^{3}$ .

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Step 3.2.1

Raise

x

$x$ to the power of

1

$1$ .

\frac{d}{d X} [(3 x^{3} + 1) {(4 \cdot 2 (x^{1} x^{3}))}^{6}]

$\frac{d}{d X} [(3 x^{3} + 1) {(4 \cdot 2 (x^{1} x^{3}))}^{6}]$

Step 3.2.2

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

\frac{d}{d X} [(3 x^{3} + 1) {(4 \cdot 2 x^{1 + 3})}^{6}]

$\frac{d}{d X} [(3 x^{3} + 1) {(4 \cdot 2 x^{1 + 3})}^{6}]$

Step 3.3

Add

$1$ and

$3$ .

$\frac{d}{d X} [(3 x^{3} + 1) {(4 \cdot 2 x^{4})}^{6}]$

Step 4

Simplify the expression.

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Step 4.1

Multiply

$4$ by

$2$ .

$\frac{d}{d X} [(3 x^{3} + 1) {(8 x^{4})}^{6}]$

Step 4.2

Apply the product rule to

$8 x^{4}$ .

$\frac{d}{d X} [(3 x^{3} + 1) (8^{6} {(x^{4})}^{6})]$

Step 4.3

Raise

$8$ to the power of

$6$ .

$\frac{d}{d X} [(3 x^{3} + 1) (262144 {(x^{4})}^{6})]$

Step 4.4

Multiply the exponents in

${(x^{4})}^{6}$ .

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Step 4.4.1

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$\frac{d}{d X} [(3 x^{3} + 1) (262144 x^{4 \cdot 6})]$

Step 4.4.2

Multiply

$4$ by

$6$ .

$\frac{d}{d X} [(3 x^{3} + 1) (262144 x^{24})]$

Step 4.5

Move

$262144$ to the left of

$3 x^{3} + 1$ .

$\frac{d}{d X} [262144 \cdot (3 x^{3} + 1) x^{24}]$

Step 5

Since

$262144 (3 x^{3} + 1) x^{24}$ is constant with respect to

$X$ , the derivative of

$262144 (3 x^{3} + 1) x^{24}$ with respect to

$X$ is

$0$ .

$0$