Calculus Examples

$y = {(x^{5} \ln (x))}^{6}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{6}$ and $g (x) = x^{5} \ln (x)$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $x^{5} \ln (x)$ .

$\frac{d}{d u} [u^{6}] \frac{d}{d x} [x^{5} \ln (x)]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 6$ .

$6 u^{5} \frac{d}{d x} [x^{5} \ln (x)]$

Step 1.3

Replace all occurrences of $u$ with $x^{5} \ln (x)$ .

$6 {(x^{5} \ln (x))}^{5} \frac{d}{d x} [x^{5} \ln (x)]$

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

$6 {(x^{5} \ln (x))}^{5} (x^{5} \frac{d}{d x} [\ln (x)] + \ln (x) \frac{d}{d x} [x^{5}])$

Step 3

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

$6 {(x^{5} \ln (x))}^{5} (x^{5} \frac{1}{x} + \ln (x) \frac{d}{d x} [x^{5}])$

Step 4

Differentiate using the Power Rule.

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Step 4.1

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

$6 {(x^{5} \ln (x))}^{5} (\frac{x^{5}}{x} + \ln (x) \frac{d}{d x} [x^{5}])$

Step 4.2

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

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Step 4.2.1

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

$6 {(x^{5} \ln (x))}^{5} (\frac{x \cdot x^{4}}{x} + \ln (x) \frac{d}{d x} [x^{5}])$

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

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

$6 {(x^{5} \ln (x))}^{5} (\frac{x \cdot x^{4}}{x^{1}} + \ln (x) \frac{d}{d x} [x^{5}])$

Step 4.2.2.2

Factor $x$ out of $x^{1}$ .

$6 {(x^{5} \ln (x))}^{5} (\frac{x \cdot x^{4}}{x \cdot 1} + \ln (x) \frac{d}{d x} [x^{5}])$

Step 4.2.2.3

Cancel the common factor.

$6 {(x^{5} \ln (x))}^{5} (\frac{x \cdot x^{4}}{x \cdot 1} + \ln (x) \frac{d}{d x} [x^{5}])$

Step 4.2.2.4

Rewrite the expression.

$6 {(x^{5} \ln (x))}^{5} (\frac{x^{4}}{1} + \ln (x) \frac{d}{d x} [x^{5}])$

Step 4.2.2.5

Divide $x^{4}$ by $1$ .

$6 {(x^{5} \ln (x))}^{5} (x^{4} + \ln (x) \frac{d}{d x} [x^{5}])$

Step 4.3

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

$6 {(x^{5} \ln (x))}^{5} (x^{4} + \ln (x) (5 x^{4}))$

Step 5

Simplify.

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Step 5.1

Apply the product rule to $x^{5} \ln (x)$ .

$6 ({(x^{5})}^{5} \ln^{5} (x)) (x^{4} + \ln (x) (5 x^{4}))$

Step 5.2

Apply the distributive property.

$6 ({(x^{5})}^{5} \ln^{5} (x)) x^{4} + 6 ({(x^{5})}^{5} \ln^{5} (x)) (\ln (x) (5 x^{4}))$

Step 5.3

Combine terms.

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Step 5.3.1

Multiply the exponents in ${(x^{5})}^{5}$ .

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Step 5.3.1.1

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

$6 (x^{5 \cdot 5} \ln^{5} (x)) x^{4} + 6 ({(x^{5})}^{5} \ln^{5} (x)) (\ln (x) (5 x^{4}))$

Step 5.3.1.2

Multiply $5$ by $5$ .

$6 (x^{25} \ln^{5} (x)) x^{4} + 6 ({(x^{5})}^{5} \ln^{5} (x)) (\ln (x) (5 x^{4}))$

Step 5.3.2

Multiply $x^{25}$ by $x^{4}$ by adding the exponents.

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Step 5.3.2.1

Move $x^{4}$ .

$6 (x^{4} x^{25}) \ln^{5} (x) + 6 ({(x^{5})}^{5} \ln^{5} (x)) (\ln (x) (5 x^{4}))$

Step 5.3.2.2

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

$6 x^{4 + 25} \ln^{5} (x) + 6 ({(x^{5})}^{5} \ln^{5} (x)) (\ln (x) (5 x^{4}))$

Step 5.3.2.3

Add $4$ and $25$ .

$6 x^{29} \ln^{5} (x) + 6 ({(x^{5})}^{5} \ln^{5} (x)) (\ln (x) (5 x^{4}))$

Step 5.3.3

Multiply the exponents in ${(x^{5})}^{5}$ .

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Step 5.3.3.1

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

$6 x^{29} \ln^{5} (x) + 6 (x^{5 \cdot 5} \ln^{5} (x)) (\ln (x) (5 x^{4}))$

Step 5.3.3.2

Multiply $5$ by $5$ .

$6 x^{29} \ln^{5} (x) + 6 (x^{25} \ln^{5} (x)) (\ln (x) (5 x^{4}))$

Step 5.3.4

Multiply $5$ by $6$ .

$6 x^{29} \ln^{5} (x) + 30 x^{25} \ln^{5} (x) (\ln (x) (x^{4}))$

Step 5.3.5

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

$6 x^{29} \ln^{5} (x) + 30 x^{4 + 25} \ln^{5} (x) \ln (x)$

Step 5.3.6

Add $4$ and $25$ .

$6 x^{29} \ln^{5} (x) + 30 x^{29} \ln^{5} (x) \ln (x)$

Step 5.3.7

Raise $\ln (x)$ to the power of $1$ .

$6 x^{29} \ln^{5} (x) + 30 x^{29} (\ln^{1} (x) \ln^{5} (x))$

Step 5.3.8

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

$6 x^{29} \ln^{5} (x) + 30 x^{29} {\ln (x)}^{1 + 5}$

Step 5.3.9

Add $1$ and $5$ .

$6 x^{29} \ln^{5} (x) + 30 x^{29} \ln^{6} (x)$