Calculus Examples

$y = x {(x^{4} + 7)}^{3}$

Step 1

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 $f (x) = x$ and $g (x) = {(x^{4} + 7)}^{3}$ .

$x \frac{d}{d x} [{(x^{4} + 7)}^{3}] + {(x^{4} + 7)}^{3} \frac{d}{d x} [x]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = x^{4} + 7$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $x^{4} + 7$ .

$x (\frac{d}{d u} [u^{3}] \frac{d}{d x} [x^{4} + 7]) + {(x^{4} + 7)}^{3} \frac{d}{d x} [x]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 3$ .

$x (3 u^{2} \frac{d}{d x} [x^{4} + 7]) + {(x^{4} + 7)}^{3} \frac{d}{d x} [x]$

Step 2.3

Replace all occurrences of $u$ with $x^{4} + 7$ .

$x (3 {(x^{4} + 7)}^{2} \frac{d}{d x} [x^{4} + 7]) + {(x^{4} + 7)}^{3} \frac{d}{d x} [x]$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $x^{4} + 7$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [7]$ .

$x (3 {(x^{4} + 7)}^{2} (\frac{d}{d x} [x^{4}] + \frac{d}{d x} [7])) + {(x^{4} + 7)}^{3} \frac{d}{d x} [x]$

Step 3.2

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

$x (3 {(x^{4} + 7)}^{2} (4 x^{3} + \frac{d}{d x} [7])) + {(x^{4} + 7)}^{3} \frac{d}{d x} [x]$

Step 3.3

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$x (3 {(x^{4} + 7)}^{2} (4 x^{3} + 0)) + {(x^{4} + 7)}^{3} \frac{d}{d x} [x]$

Step 3.4

Simplify the expression.

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Step 3.4.1

Add $4 x^{3}$ and $0$ .

$x (3 {(x^{4} + 7)}^{2} (4 x^{3})) + {(x^{4} + 7)}^{3} \frac{d}{d x} [x]$

Step 3.4.2

Multiply $4$ by $3$ .

$x (12 {(x^{4} + 7)}^{2} x^{3}) + {(x^{4} + 7)}^{3} \frac{d}{d x} [x]$

Step 4

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

$x^{3} x^{1} (12 {(x^{4} + 7)}^{2}) + {(x^{4} + 7)}^{3} \frac{d}{d x} [x]$

Step 5

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

$x^{3 + 1} (12 {(x^{4} + 7)}^{2}) + {(x^{4} + 7)}^{3} \frac{d}{d x} [x]$

Step 6

Add $3$ and $1$ .

$x^{4} (12 {(x^{4} + 7)}^{2}) + {(x^{4} + 7)}^{3} \frac{d}{d x} [x]$

Step 7

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

$x^{4} (12 {(x^{4} + 7)}^{2}) + {(x^{4} + 7)}^{3} \cdot 1$

Step 8

Multiply ${(x^{4} + 7)}^{3}$ by $1$ .

$x^{4} (12 {(x^{4} + 7)}^{2}) + {(x^{4} + 7)}^{3}$

Step 9

Simplify.

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Step 9.1

Factor ${(x^{4} + 7)}^{2}$ out of $x^{4} (12 {(x^{4} + 7)}^{2}) + {(x^{4} + 7)}^{3}$ .

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Step 9.1.1

Factor ${(x^{4} + 7)}^{2}$ out of $x^{4} (12 {(x^{4} + 7)}^{2})$ .

${(x^{4} + 7)}^{2} (x^{4} \cdot (12)) + {(x^{4} + 7)}^{3}$

Step 9.1.2

Factor ${(x^{4} + 7)}^{2}$ out of ${(x^{4} + 7)}^{3}$ .

${(x^{4} + 7)}^{2} (x^{4} \cdot (12)) + {(x^{4} + 7)}^{2} (x^{4} + 7)$

Step 9.1.3

Factor ${(x^{4} + 7)}^{2}$ out of ${(x^{4} + 7)}^{2} (x^{4} \cdot (12)) + {(x^{4} + 7)}^{2} (x^{4} + 7)$ .

${(x^{4} + 7)}^{2} (x^{4} \cdot (12) + x^{4} + 7)$

Step 9.2

Combine terms.

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Step 9.2.1

Move $12$ to the left of $x^{4}$ .

${(x^{4} + 7)}^{2} (12 \cdot x^{4} + x^{4} + 7)$

Step 9.2.2

Add $12 x^{4}$ and $x^{4}$ .

${(x^{4} + 7)}^{2} (13 x^{4} + 7)$

Step 9.3

Rewrite ${(x^{4} + 7)}^{2}$ as $(x^{4} + 7) (x^{4} + 7)$ .

$(x^{4} + 7) (x^{4} + 7) (13 x^{4} + 7)$

Step 9.4

Expand $(x^{4} + 7) (x^{4} + 7)$ using the FOIL Method.

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Step 9.4.1

Apply the distributive property.

$(x^{4} (x^{4} + 7) + 7 (x^{4} + 7)) (13 x^{4} + 7)$

Step 9.4.2

Apply the distributive property.

$(x^{4} x^{4} + x^{4} \cdot 7 + 7 (x^{4} + 7)) (13 x^{4} + 7)$

Step 9.4.3

Apply the distributive property.

$(x^{4} x^{4} + x^{4} \cdot 7 + 7 x^{4} + 7 \cdot 7) (13 x^{4} + 7)$

Step 9.5

Simplify and combine like terms.

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Step 9.5.1

Simplify each term.

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Step 9.5.1.1

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

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Step 9.5.1.1.1

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

$(x^{4 + 4} + x^{4} \cdot 7 + 7 x^{4} + 7 \cdot 7) (13 x^{4} + 7)$

Step 9.5.1.1.2

Add $4$ and $4$ .

$(x^{8} + x^{4} \cdot 7 + 7 x^{4} + 7 \cdot 7) (13 x^{4} + 7)$

Step 9.5.1.2

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

$(x^{8} + 7 \cdot x^{4} + 7 x^{4} + 7 \cdot 7) (13 x^{4} + 7)$

Step 9.5.1.3

Multiply $7$ by $7$ .

$(x^{8} + 7 x^{4} + 7 x^{4} + 49) (13 x^{4} + 7)$

Step 9.5.2

Add $7 x^{4}$ and $7 x^{4}$ .

$(x^{8} + 14 x^{4} + 49) (13 x^{4} + 7)$

Step 9.6

Expand $(x^{8} + 14 x^{4} + 49) (13 x^{4} + 7)$ by multiplying each term in the first expression by each term in the second expression.

$x^{8} (13 x^{4}) + x^{8} \cdot 7 + 14 x^{4} (13 x^{4}) + 14 x^{4} \cdot 7 + 49 (13 x^{4}) + 49 \cdot 7$

Step 9.7

Simplify each term.

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Step 9.7.1

Rewrite using the commutative property of multiplication.

$13 x^{8} x^{4} + x^{8} \cdot 7 + 14 x^{4} (13 x^{4}) + 14 x^{4} \cdot 7 + 49 (13 x^{4}) + 49 \cdot 7$

Step 9.7.2

Multiply $x^{8}$ by $x^{4}$ by adding the exponents.

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Step 9.7.2.1

Move $x^{4}$ .

$13 (x^{4} x^{8}) + x^{8} \cdot 7 + 14 x^{4} (13 x^{4}) + 14 x^{4} \cdot 7 + 49 (13 x^{4}) + 49 \cdot 7$

Step 9.7.2.2

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

$13 x^{4 + 8} + x^{8} \cdot 7 + 14 x^{4} (13 x^{4}) + 14 x^{4} \cdot 7 + 49 (13 x^{4}) + 49 \cdot 7$

Step 9.7.2.3

Add $4$ and $8$ .

$13 x^{12} + x^{8} \cdot 7 + 14 x^{4} (13 x^{4}) + 14 x^{4} \cdot 7 + 49 (13 x^{4}) + 49 \cdot 7$

Step 9.7.3

Move $7$ to the left of $x^{8}$ .

$13 x^{12} + 7 \cdot x^{8} + 14 x^{4} (13 x^{4}) + 14 x^{4} \cdot 7 + 49 (13 x^{4}) + 49 \cdot 7$

Step 9.7.4

Rewrite using the commutative property of multiplication.

$13 x^{12} + 7 x^{8} + 14 \cdot 13 x^{4} x^{4} + 14 x^{4} \cdot 7 + 49 (13 x^{4}) + 49 \cdot 7$

Step 9.7.5

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

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Step 9.7.5.1

Move $x^{4}$ .

$13 x^{12} + 7 x^{8} + 14 \cdot 13 (x^{4} x^{4}) + 14 x^{4} \cdot 7 + 49 (13 x^{4}) + 49 \cdot 7$

Step 9.7.5.2

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

$13 x^{12} + 7 x^{8} + 14 \cdot 13 x^{4 + 4} + 14 x^{4} \cdot 7 + 49 (13 x^{4}) + 49 \cdot 7$

Step 9.7.5.3

Add $4$ and $4$ .

$13 x^{12} + 7 x^{8} + 14 \cdot 13 x^{8} + 14 x^{4} \cdot 7 + 49 (13 x^{4}) + 49 \cdot 7$

Step 9.7.6

Multiply $14$ by $13$ .

$13 x^{12} + 7 x^{8} + 182 x^{8} + 14 x^{4} \cdot 7 + 49 (13 x^{4}) + 49 \cdot 7$

Step 9.7.7

Multiply $7$ by $14$ .

$13 x^{12} + 7 x^{8} + 182 x^{8} + 98 x^{4} + 49 (13 x^{4}) + 49 \cdot 7$

Step 9.7.8

Multiply $13$ by $49$ .

$13 x^{12} + 7 x^{8} + 182 x^{8} + 98 x^{4} + 637 x^{4} + 49 \cdot 7$

Step 9.7.9

Multiply $49$ by $7$ .

$13 x^{12} + 7 x^{8} + 182 x^{8} + 98 x^{4} + 637 x^{4} + 343$

Step 9.8

Add $7 x^{8}$ and $182 x^{8}$ .

$13 x^{12} + 189 x^{8} + 98 x^{4} + 637 x^{4} + 343$

Step 9.9

Add $98 x^{4}$ and $637 x^{4}$ .

$13 x^{12} + 189 x^{8} + 735 x^{4} + 343$