Calculus Examples

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

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{5}$ and $g (x) = x^{3} \ln (x)$ .

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To apply the Chain Rule, set $u$ as $x^{3} \ln (x)$ .

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

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

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

Replace all occurrences of $u$ with $x^{3} \ln (x)$ .

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

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

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

Step 3

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

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

Step 4

Differentiate using the Power Rule.

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

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

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

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

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

Cancel the common factors.

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

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

Step 5

Simplify.

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Apply the product rule to $x^{3} \ln (x)$ .

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

Apply the distributive property.

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

Combine terms.

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Multiply the exponents in ${(x^{3})}^{4}$ .

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

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

Multiply $3$ by $4$ .

$5 (x^{12} \ln^{4} (x)) x^{2} + 5 ({(x^{3})}^{4} \ln^{4} (x)) (\ln (x) (3 x^{2}))$

Multiply $x^{12}$ by $x^{2}$ by adding the exponents.

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Move $x^{2}$ .

$5 (x^{2} x^{12}) \ln^{4} (x) + 5 ({(x^{3})}^{4} \ln^{4} (x)) (\ln (x) (3 x^{2}))$

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

$5 x^{2 + 12} \ln^{4} (x) + 5 ({(x^{3})}^{4} \ln^{4} (x)) (\ln (x) (3 x^{2}))$

Add $2$ and $12$ .

$5 x^{14} \ln^{4} (x) + 5 ({(x^{3})}^{4} \ln^{4} (x)) (\ln (x) (3 x^{2}))$

Multiply the exponents in ${(x^{3})}^{4}$ .

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

$5 x^{14} \ln^{4} (x) + 5 (x^{3 \cdot 4} \ln^{4} (x)) (\ln (x) (3 x^{2}))$

Multiply $3$ by $4$ .

$5 x^{14} \ln^{4} (x) + 5 (x^{12} \ln^{4} (x)) (\ln (x) (3 x^{2}))$

Multiply $3$ by $5$ .

$5 x^{14} \ln^{4} (x) + 15 x^{12} \ln^{4} (x) (\ln (x) (x^{2}))$

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

$5 x^{14} \ln^{4} (x) + 15 x^{2 + 12} \ln^{4} (x) \ln (x)$

Add $2$ and $12$ .

$5 x^{14} \ln^{4} (x) + 15 x^{14} \ln^{4} (x) \ln (x)$

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

$5 x^{14} \ln^{4} (x) + 15 x^{14} (\ln^{1} (x) \ln^{4} (x))$

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

$5 x^{14} \ln^{4} (x) + 15 x^{14} {\ln (x)}^{1 + 4}$

Add $1$ and $4$ .

$5 x^{14} \ln^{4} (x) + 15 x^{14} \ln^{5} (x)$