Calculus Examples

$y = 3 (4 x + \ln ({(x^{2})}^{2}))$

Step 1

Differentiate.

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Step 1.1

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 1.1.2

Multiply $2$ by $2$ .

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

Multiply $4$ by $1$ .

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

Step 2

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

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Step 2.1

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

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

Step 2.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

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

Step 2.3

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

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Step 2.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.3.2

Multiply $2$ by $2$ .

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

Step 3

Differentiate using the Power Rule.

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Step 3.1

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

$3 (4 + \frac{1}{x^{4}} (4 x^{3}))$

Step 3.2

Simplify terms.

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Step 3.2.1

Combine $4$ and $\frac{1}{x^{4}}$ .

$3 (4 + \frac{4}{x^{4}} x^{3})$

Step 3.2.2

Combine $\frac{4}{x^{4}}$ and $x^{3}$ .

$3 (4 + \frac{4 x^{3}}{x^{4}})$

Step 3.2.3

Cancel the common factor of $x^{3}$ and $x^{4}$ .

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Step 3.2.3.1

Factor $x^{3}$ out of $4 x^{3}$ .

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

Step 3.2.3.2

Cancel the common factors.

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Step 3.2.3.2.1

Factor $x^{3}$ out of $x^{4}$ .

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

Step 3.2.3.2.2

Cancel the common factor.

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

Step 3.2.3.2.3

Rewrite the expression.

$3 (4 + \frac{4}{x})$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$3 \cdot 4 + 3 \frac{4}{x}$

Step 4.2

Combine terms.

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Step 4.2.1

Multiply $3$ by $4$ .

$12 + 3 \frac{4}{x}$

Step 4.2.2

Combine $3$ and $\frac{4}{x}$ .

$12 + \frac{3 \cdot 4}{x}$

Step 4.2.3

Multiply $3$ by $4$ .

$12 + \frac{12}{x}$

Step 4.3

Reorder terms.

$\frac{12}{x} + 12$