Calculus Examples

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

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = x^{2} (x + 1)$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $x^{2} (x + 1)$ .

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

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u$ with $x^{2} (x + 1)$ .

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

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 $f (x) = x^{2}$ and $g (x) = x + 1$ .

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

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $x + 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [1]$ .

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

Step 3.2

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

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

Step 3.3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{1}{x^{2} (x + 1)} (x^{2} (1 + 0) + (x + 1) \frac{d}{d x} [x^{2}])$

Step 3.4

Simplify the expression.

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Step 3.4.1

Add $1$ and $0$ .

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

Step 3.4.2

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

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

Step 3.5

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

$\frac{1}{x^{2} (x + 1)} (x^{2} + (x + 1) (2 x))$

Step 3.6

Move $2$ to the left of $x + 1$ .

$\frac{1}{x^{2} (x + 1)} (x^{2} + 2 \cdot (x + 1) x)$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$\frac{1}{x^{2} x + x^{2} \cdot 1} (x^{2} + 2 (x + 1) x)$

Step 4.2

Apply the distributive property.

$\frac{1}{x^{2} x + x^{2} \cdot 1} (x^{2} + (2 x + 2 \cdot 1) x)$

Step 4.3

Apply the distributive property.

$\frac{1}{x^{2} x + x^{2} \cdot 1} (x^{2} + 2 x \cdot x + 2 \cdot 1 x)$

Step 4.4

Combine terms.

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Step 4.4.1

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

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Step 4.4.1.1

Multiply $x^{2}$ by $x$ .

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Step 4.4.1.1.1

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

$\frac{1}{x^{2} x^{1} + x^{2} \cdot 1} (x^{2} + 2 x \cdot x + 2 \cdot 1 x)$

Step 4.4.1.1.2

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

$\frac{1}{x^{2 + 1} + x^{2} \cdot 1} (x^{2} + 2 x \cdot x + 2 \cdot 1 x)$

Step 4.4.1.2

Add $2$ and $1$ .

$\frac{1}{x^{3} + x^{2} \cdot 1} (x^{2} + 2 x \cdot x + 2 \cdot 1 x)$

Step 4.4.2

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

$\frac{1}{x^{3} + x^{2}} (x^{2} + 2 x \cdot x + 2 \cdot 1 x)$

Step 4.4.3

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

$\frac{1}{x^{3} + x^{2}} (x^{2} + 2 (x^{1} x) + 2 \cdot 1 x)$

Step 4.4.4

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

$\frac{1}{x^{3} + x^{2}} (x^{2} + 2 (x^{1} x^{1}) + 2 \cdot 1 x)$

Step 4.4.5

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

$\frac{1}{x^{3} + x^{2}} (x^{2} + 2 x^{1 + 1} + 2 \cdot 1 x)$

Step 4.4.6

Add $1$ and $1$ .

$\frac{1}{x^{3} + x^{2}} (x^{2} + 2 x^{2} + 2 \cdot 1 x)$

Step 4.4.7

Multiply $2$ by $1$ .

$\frac{1}{x^{3} + x^{2}} (x^{2} + 2 x^{2} + 2 x)$

Step 4.4.8

Add $x^{2}$ and $2 x^{2}$ .

$\frac{1}{x^{3} + x^{2}} (3 x^{2} + 2 x)$

Step 4.5

Reorder the factors of $\frac{1}{x^{3} + x^{2}} (3 x^{2} + 2 x)$ .

$(3 x^{2} + 2 x) \frac{1}{x^{3} + x^{2}}$

Step 4.6

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

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Step 4.6.1

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

$(3 x^{2} + 2 x) \frac{1}{x^{2} x + x^{2}}$

Step 4.6.2

Multiply by $1$ .

$(3 x^{2} + 2 x) \frac{1}{x^{2} x + x^{2} \cdot 1}$

Step 4.6.3

Factor $x^{2}$ out of $x^{2} x + x^{2} \cdot 1$ .

$(3 x^{2} + 2 x) \frac{1}{x^{2} (x + 1)}$

Step 4.7

Multiply $3 x^{2} + 2 x$ by $\frac{1}{x^{2} (x + 1)}$ .

$\frac{3 x^{2} + 2 x}{x^{2} (x + 1)}$

Step 4.8

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

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Step 4.8.1

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

$\frac{x (3 x) + 2 x}{x^{2} (x + 1)}$

Step 4.8.2

Factor $x$ out of $2 x$ .

$\frac{x (3 x) + x \cdot 2}{x^{2} (x + 1)}$

Step 4.8.3

Factor $x$ out of $x (3 x) + x \cdot 2$ .

$\frac{x (3 x + 2)}{x^{2} (x + 1)}$

Step 4.9

Cancel the common factors.

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Step 4.9.1

Factor $x$ out of $x^{2} (x + 1)$ .

$\frac{x (3 x + 2)}{x (x (x + 1))}$

Step 4.9.2

Cancel the common factor.

$\frac{x (3 x + 2)}{x (x (x + 1))}$

Step 4.9.3

Rewrite the expression.

$\frac{3 x + 2}{x (x + 1)}$