Algebra Examples

$x^{4} {(a - 2 x^{3})}^{2}$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d a} [f (a) g (a)]$ is $f (a) \frac{d}{d a} [g (a)] + g (a) \frac{d}{d a} [f (a)]$ where $f (a) = x^{4}$ and $g (a) = {(a - 2 x^{3})}^{2}$ .

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

Step 2

Rewrite ${(a - 2 x^{3})}^{2}$ as $(a - 2 x^{3}) (a - 2 x^{3})$ .

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

Step 3

Expand $(a - 2 x^{3}) (a - 2 x^{3})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $a$ by $a$ .

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

Step 4.1.2

Rewrite using the commutative property of multiplication.

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

Step 4.1.3

Rewrite using the commutative property of multiplication.

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

Step 4.1.4

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

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Step 4.1.4.1

Move $x^{3}$ .

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

Step 4.1.4.2

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

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

Step 4.1.4.3

Add $3$ and $3$ .

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

Step 4.1.5

Multiply $- 2$ by $- 2$ .

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

Step 4.2

Subtract $2 x^{3} a$ from $- 2 a x^{3}$ .

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Step 4.2.1

Move $x^{3}$ .

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

Step 4.2.2

Subtract $2 a x^{3}$ from $- 2 a x^{3}$ .

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

Step 5

By the Sum Rule, the derivative of $a^{2} - 4 a x^{3} + 4 x^{6}$ with respect to $a$ is $\frac{d}{d a} [a^{2}] + \frac{d}{d a} [- 4 a x^{3}] + \frac{d}{d a} [4 x^{6}]$ .

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Multiply $- 4$ by $1$ .

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

Step 10

Since $4 x^{6}$ is constant with respect to $a$ , the derivative of $4 x^{6}$ with respect to $a$ is $0$ .

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

Step 11

Simplify the expression.

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Step 11.1

Add $2 a - 4 x^{3}$ and $0$ .

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

Step 11.2

Reorder terms.

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

Step 12

Since $x^{4}$ is constant with respect to $a$ , the derivative of $x^{4}$ with respect to $a$ is $0$ .

$x^{4} (- 4 x^{3} + 2 a) + {(a - 2 x^{3})}^{2} \cdot 0$

Step 13

Simplify.

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Step 13.1

Apply the distributive property.

$x^{4} (- 4 x^{3}) + x^{4} (2 a) + {(a - 2 x^{3})}^{2} \cdot 0$

Step 13.2

Combine terms.

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Step 13.2.1

Multiply $x^{4}$ by $x^{3}$ by adding the exponents.

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Step 13.2.1.1

Move $x^{3}$ .

$x^{3} x^{4} \cdot - 4 + x^{4} (2 a) + {(a - 2 x^{3})}^{2} \cdot 0$

Step 13.2.1.2

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

$x^{3 + 4} \cdot - 4 + x^{4} (2 a) + {(a - 2 x^{3})}^{2} \cdot 0$

Step 13.2.1.3

Add $3$ and $4$ .

$x^{7} \cdot - 4 + x^{4} (2 a) + {(a - 2 x^{3})}^{2} \cdot 0$

Step 13.2.2

Move $- 4$ to the left of $x^{7}$ .

$- 4 \cdot x^{7} + x^{4} (2 a) + {(a - 2 x^{3})}^{2} \cdot 0$

Step 13.2.3

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

$- 4 x^{7} + 2 \cdot x^{4} a + {(a - 2 x^{3})}^{2} \cdot 0$

Step 13.2.4

Multiply ${(a - 2 x^{3})}^{2}$ by $0$ .

$- 4 x^{7} + 2 x^{4} a + 0$

Step 13.2.5

Add $- 4 x^{7} + 2 x^{4} a$ and $0$ .

$- 4 x^{7} + 2 x^{4} a$