Calculus Examples

${(3 x^{4} + 8)}^{2}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = 3 x^{4} + 8$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $3 x^{4} + 8$ .

$\frac{d}{d u} [u^{2}] \frac{d}{d x} [3 x^{4} + 8]$

Step 1.2

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

$2 u \frac{d}{d x} [3 x^{4} + 8]$

Step 1.3

Replace all occurrences of $u$ with $3 x^{4} + 8$ .

$2 (3 x^{4} + 8) \frac{d}{d x} [3 x^{4} + 8]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $3 x^{4} + 8$ with respect to $x$ is $\frac{d}{d x} [3 x^{4}] + \frac{d}{d x} [8]$ .

$2 (3 x^{4} + 8) (\frac{d}{d x} [3 x^{4}] + \frac{d}{d x} [8])$

Step 2.2

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

$2 (3 x^{4} + 8) (3 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [8])$

Step 2.3

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

$2 (3 x^{4} + 8) (3 (4 x^{3}) + \frac{d}{d x} [8])$

Step 2.4

Multiply $4$ by $3$ .

$2 (3 x^{4} + 8) (12 x^{3} + \frac{d}{d x} [8])$

Step 2.5

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$2 (3 x^{4} + 8) (12 x^{3} + 0)$

Step 2.6

Simplify the expression.

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Step 2.6.1

Add $12 x^{3}$ and $0$ .

$2 (3 x^{4} + 8) (12 x^{3})$

Step 2.6.2

Multiply $12$ by $2$ .

$24 (3 x^{4} + 8) x^{3}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$(24 (3 x^{4}) + 24 \cdot 8) x^{3}$

Step 3.2

Apply the distributive property.

$24 (3 x^{4}) x^{3} + 24 \cdot 8 x^{3}$

Step 3.3

Combine terms.

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Step 3.3.1

Multiply $3$ by $24$ .

$72 x^{4} x^{3} + 24 \cdot 8 x^{3}$

Step 3.3.2

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

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Step 3.3.2.1

Move $x^{3}$ .

$72 (x^{3} x^{4}) + 24 \cdot 8 x^{3}$

Step 3.3.2.2

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

$72 x^{3 + 4} + 24 \cdot 8 x^{3}$

Step 3.3.2.3

Add $3$ and $4$ .

$72 x^{7} + 24 \cdot 8 x^{3}$

Step 3.3.3

Multiply $24$ by $8$ .

$72 x^{7} + 192 x^{3}$