Calculus Examples

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

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} ({(x^{2} \ln (x))}^{4})$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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Step 3.1.1

To apply the Chain Rule, set $u$ as $x^{2} \ln (x)$ .

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

Step 3.1.2

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

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

Step 3.1.3

Replace all occurrences of $u$ with $x^{2} \ln (x)$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Differentiate using the Power Rule.

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Step 3.4.1

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

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

Step 3.4.2

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

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Step 3.4.2.1

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

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

Step 3.4.2.2

Cancel the common factors.

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Step 3.4.2.2.1

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

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

Step 3.4.2.2.2

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

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

Step 3.4.2.2.3

Cancel the common factor.

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

Step 3.4.2.2.4

Rewrite the expression.

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

Step 3.4.2.2.5

Divide $x$ by $1$ .

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

Step 3.4.3

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

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

Step 3.5

Simplify.

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Step 3.5.1

Apply the product rule to $x^{2} \ln (x)$ .

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

Step 3.5.2

Apply the distributive property.

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

Step 3.5.3

Combine terms.

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Step 3.5.3.1

Multiply the exponents in ${(x^{2})}^{3}$ .

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Step 3.5.3.1.1

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

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

Step 3.5.3.1.2

Multiply $2$ by $3$ .

$4 (x^{6} \ln^{3} (x)) x + 4 ({(x^{2})}^{3} \ln^{3} (x)) (\ln (x) (2 x))$

Step 3.5.3.2

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

$4 (x^{1} x^{6}) \ln^{3} (x) + 4 ({(x^{2})}^{3} \ln^{3} (x)) (\ln (x) (2 x))$

Step 3.5.3.3

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

$4 x^{1 + 6} \ln^{3} (x) + 4 ({(x^{2})}^{3} \ln^{3} (x)) (\ln (x) (2 x))$

Step 3.5.3.4

Add $1$ and $6$ .

$4 x^{7} \ln^{3} (x) + 4 ({(x^{2})}^{3} \ln^{3} (x)) (\ln (x) (2 x))$

Step 3.5.3.5

Multiply the exponents in ${(x^{2})}^{3}$ .

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Step 3.5.3.5.1

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

$4 x^{7} \ln^{3} (x) + 4 (x^{2 \cdot 3} \ln^{3} (x)) (\ln (x) (2 x))$

Step 3.5.3.5.2

Multiply $2$ by $3$ .

$4 x^{7} \ln^{3} (x) + 4 (x^{6} \ln^{3} (x)) (\ln (x) (2 x))$

Step 3.5.3.6

Multiply $2$ by $4$ .

$4 x^{7} \ln^{3} (x) + 8 x^{6} \ln^{3} (x) (\ln (x) (x))$

Step 3.5.3.7

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

$4 x^{7} \ln^{3} (x) + 8 (x^{1} x^{6}) \ln^{3} (x) \ln (x)$

Step 3.5.3.8

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

$4 x^{7} \ln^{3} (x) + 8 x^{1 + 6} \ln^{3} (x) \ln (x)$

Step 3.5.3.9

Add $1$ and $6$ .

$4 x^{7} \ln^{3} (x) + 8 x^{7} \ln^{3} (x) \ln (x)$

Step 3.5.3.10

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

$4 x^{7} \ln^{3} (x) + 8 x^{7} (\ln^{1} (x) \ln^{3} (x))$

Step 3.5.3.11

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

$4 x^{7} \ln^{3} (x) + 8 x^{7} {\ln (x)}^{1 + 3}$

Step 3.5.3.12

Add $1$ and $3$ .

$4 x^{7} \ln^{3} (x) + 8 x^{7} \ln^{4} (x)$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = 4 x^{7} \ln^{3} (x) + 8 x^{7} \ln^{4} (x)$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = 4 x^{7} \ln^{3} (x) + 8 x^{7} \ln^{4} (x)$