Calculus Examples

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

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.

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

Step 3

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

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Step 3.1

Move $x$ .

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

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 $0$ .

$0$