Calculus Examples

$y = 4 x^{3} \ln (2 x^{5})$

Step 1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{3} \ln (2 x^{5})$ with respect to $x$ is $4 \frac{d}{d x} [x^{3} \ln (2 x^{5})]$ .

$4 \frac{d}{d x} [x^{3} \ln (2 x^{5})]$

Step 2

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 $f (x) = x^{3}$ and $g (x) = \ln (2 x^{5})$ .

$4 (x^{3} \frac{d}{d x} [\ln (2 x^{5})] + \ln (2 x^{5}) \frac{d}{d x} [x^{3}])$

Step 3

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 $g (x) = 2 x^{5}$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $2 x^{5}$ .

$4 (x^{3} (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [2 x^{5}]) + \ln (2 x^{5}) \frac{d}{d x} [x^{3}])$

Step 3.2

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

$4 (x^{3} (\frac{1}{u} \frac{d}{d x} [2 x^{5}]) + \ln (2 x^{5}) \frac{d}{d x} [x^{3}])$

Step 3.3

Replace all occurrences of $u$ with $2 x^{5}$ .

$4 (x^{3} (\frac{1}{2 x^{5}} \frac{d}{d x} [2 x^{5}]) + \ln (2 x^{5}) \frac{d}{d x} [x^{3}])$

Step 4

Differentiate.

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Step 4.1

Combine $\frac{1}{2 x^{5}}$ and $x^{3}$ .

$4 (\frac{x^{3}}{2 x^{5}} \frac{d}{d x} [2 x^{5}] + \ln (2 x^{5}) \frac{d}{d x} [x^{3}])$

Step 4.2

Cancel the common factor of $x^{3}$ and $x^{5}$ .

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Step 4.2.1

Multiply by $1$ .

$4 (\frac{x^{3} \cdot 1}{2 x^{5}} \frac{d}{d x} [2 x^{5}] + \ln (2 x^{5}) \frac{d}{d x} [x^{3}])$

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

Factor $x^{3}$ out of $2 x^{5}$ .

$4 (\frac{x^{3} \cdot 1}{x^{3} (2 x^{2})} \frac{d}{d x} [2 x^{5}] + \ln (2 x^{5}) \frac{d}{d x} [x^{3}])$

Step 4.2.2.2

Cancel the common factor.

$4 (\frac{x^{3} \cdot 1}{x^{3} (2 x^{2})} \frac{d}{d x} [2 x^{5}] + \ln (2 x^{5}) \frac{d}{d x} [x^{3}])$

Step 4.2.2.3

Rewrite the expression.

$4 (\frac{1}{2 x^{2}} \frac{d}{d x} [2 x^{5}] + \ln (2 x^{5}) \frac{d}{d x} [x^{3}])$

Step 4.3

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{5}$ with respect to $x$ is $2 \frac{d}{d x} [x^{5}]$ .

$4 (\frac{1}{2 x^{2}} (2 \frac{d}{d x} [x^{5}]) + \ln (2 x^{5}) \frac{d}{d x} [x^{3}])$

Step 4.4

Simplify terms.

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Step 4.4.1

Combine $2$ and $\frac{1}{2 x^{2}}$ .

$4 (\frac{2}{2 x^{2}} \frac{d}{d x} [x^{5}] + \ln (2 x^{5}) \frac{d}{d x} [x^{3}])$

Step 4.4.2

Cancel the common factor of $2$ .

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Step 4.4.2.1

Cancel the common factor.

$4 (\frac{2}{2 x^{2}} \frac{d}{d x} [x^{5}] + \ln (2 x^{5}) \frac{d}{d x} [x^{3}])$

Step 4.4.2.2

Rewrite the expression.

$4 (\frac{1}{x^{2}} \frac{d}{d x} [x^{5}] + \ln (2 x^{5}) \frac{d}{d x} [x^{3}])$

Step 4.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$4 (\frac{1}{x^{2}} (5 x^{4}) + \ln (2 x^{5}) \frac{d}{d x} [x^{3}])$

Step 4.6

Simplify terms.

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Step 4.6.1

Combine $5$ and $\frac{1}{x^{2}}$ .

$4 (\frac{5}{x^{2}} x^{4} + \ln (2 x^{5}) \frac{d}{d x} [x^{3}])$

Step 4.6.2

Combine $\frac{5}{x^{2}}$ and $x^{4}$ .

$4 (\frac{5 x^{4}}{x^{2}} + \ln (2 x^{5}) \frac{d}{d x} [x^{3}])$

Step 4.6.3

Cancel the common factor of $x^{4}$ and $x^{2}$ .

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Step 4.6.3.1

Factor $x^{2}$ out of $5 x^{4}$ .

$4 (\frac{x^{2} (5 x^{2})}{x^{2}} + \ln (2 x^{5}) \frac{d}{d x} [x^{3}])$

Step 4.6.3.2

Cancel the common factors.

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Step 4.6.3.2.1

Multiply by $1$ .

$4 (\frac{x^{2} (5 x^{2})}{x^{2} \cdot 1} + \ln (2 x^{5}) \frac{d}{d x} [x^{3}])$

Step 4.6.3.2.2

Cancel the common factor.

$4 (\frac{x^{2} (5 x^{2})}{x^{2} \cdot 1} + \ln (2 x^{5}) \frac{d}{d x} [x^{3}])$

Step 4.6.3.2.3

Rewrite the expression.

$4 (\frac{5 x^{2}}{1} + \ln (2 x^{5}) \frac{d}{d x} [x^{3}])$

Step 4.6.3.2.4

Divide $5 x^{2}$ by $1$ .

$4 (5 x^{2} + \ln (2 x^{5}) \frac{d}{d x} [x^{3}])$

Step 4.7

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

$4 (5 x^{2} + \ln (2 x^{5}) (3 x^{2}))$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$4 (5 x^{2}) + 4 (\ln (2 x^{5}) (3 x^{2}))$

Step 5.2

Combine terms.

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Step 5.2.1

Multiply $5$ by $4$ .

$20 x^{2} + 4 (\ln (2 x^{5}) (3 x^{2}))$

Step 5.2.2

Multiply $3$ by $4$ .

$20 x^{2} + 12 \ln (2 x^{5}) x^{2}$

Step 5.3

Reorder terms.

$20 x^{2} + 12 x^{2} \ln (2 x^{5})$