Calculus Examples

$y = x \ln^{2} (11 x)$

Step 1

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 $f (x) = x$ and $g (x) = \ln^{2} (11 x)$ .

$x \frac{d}{d x} [\ln^{2} (11 x)] + \ln^{2} (11 x) \frac{d}{d x} [x]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \ln (11 x)$ .

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Step 2.1

To apply the Chain Rule, set $u_{1}$ as $\ln (11 x)$ .

$x (\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d x} [\ln (11 x)]) + \ln^{2} (11 x) \frac{d}{d x} [x]$

Step 2.2

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

$x (2 u_{1} \frac{d}{d x} [\ln (11 x)]) + \ln^{2} (11 x) \frac{d}{d x} [x]$

Step 2.3

Replace all occurrences of $u_{1}$ with $\ln (11 x)$ .

$x (2 \ln (11 x) \frac{d}{d x} [\ln (11 x)]) + \ln^{2} (11 x) \frac{d}{d x} [x]$

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 $g (x) = 11 x$ .

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Step 3.1

To apply the Chain Rule, set $u_{2}$ as $11 x$ .

$x (2 \ln (11 x) (\frac{d}{d u_{2}} [\ln (u_{2})] \frac{d}{d x} [11 x])) + \ln^{2} (11 x) \frac{d}{d x} [x]$

Step 3.2

The derivative of $\ln (u_{2})$ with respect to $u_{2}$ is $\frac{1}{u_{2}}$ .

$x (2 \ln (11 x) (\frac{1}{u_{2}} \frac{d}{d x} [11 x])) + \ln^{2} (11 x) \frac{d}{d x} [x]$

Step 3.3

Replace all occurrences of $u_{2}$ with $11 x$ .

$x (2 \ln (11 x) (\frac{1}{11 x} \frac{d}{d x} [11 x])) + \ln^{2} (11 x) \frac{d}{d x} [x]$

Step 4

Differentiate.

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Step 4.1

Combine $\frac{1}{11 x}$ and $2$ .

$x (\frac{2}{11 x} \ln (11 x) \frac{d}{d x} [11 x]) + \ln^{2} (11 x) \frac{d}{d x} [x]$

Step 4.2

Simplify terms.

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Step 4.2.1

Combine $\frac{2}{11 x}$ and $\ln (11 x)$ .

$x (\frac{2 \ln (11 x)}{11 x} \frac{d}{d x} [11 x]) + \ln^{2} (11 x) \frac{d}{d x} [x]$

Step 4.2.2

Combine $\frac{2 \ln (11 x)}{11 x}$ and $x$ .

$\frac{2 \ln (11 x) x}{11 x} \frac{d}{d x} [11 x] + \ln^{2} (11 x) \frac{d}{d x} [x]$

Step 4.2.3

Cancel the common factor of $x$ .

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Step 4.2.3.1

Cancel the common factor.

$\frac{2 \ln (11 x) x}{11 x} \frac{d}{d x} [11 x] + \ln^{2} (11 x) \frac{d}{d x} [x]$

Step 4.2.3.2

Rewrite the expression.

$\frac{2 \ln (11 x)}{11} \frac{d}{d x} [11 x] + \ln^{2} (11 x) \frac{d}{d x} [x]$

Step 4.3

Since $11$ is constant with respect to $x$ , the derivative of $11 x$ with respect to $x$ is $11 \frac{d}{d x} [x]$ .

$\frac{2 \ln (11 x)}{11} (11 \frac{d}{d x} [x]) + \ln^{2} (11 x) \frac{d}{d x} [x]$

Step 4.4

Simplify terms.

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Step 4.4.1

Combine $11$ and $\frac{2 \ln (11 x)}{11}$ .

$\frac{11 (2 \ln (11 x))}{11} \frac{d}{d x} [x] + \ln^{2} (11 x) \frac{d}{d x} [x]$

Step 4.4.2

Multiply $2$ by $11$ .

$\frac{22 \ln (11 x)}{11} \frac{d}{d x} [x] + \ln^{2} (11 x) \frac{d}{d x} [x]$

Step 4.4.3

Cancel the common factor of $22$ and $11$ .

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Step 4.4.3.1

Factor $11$ out of $22 \ln (11 x)$ .

$\frac{11 (2 \ln (11 x))}{11} \frac{d}{d x} [x] + \ln^{2} (11 x) \frac{d}{d x} [x]$

Step 4.4.3.2

Cancel the common factors.

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Step 4.4.3.2.1

Factor $11$ out of $11$ .

$\frac{11 (2 \ln (11 x))}{11 (1)} \frac{d}{d x} [x] + \ln^{2} (11 x) \frac{d}{d x} [x]$

Step 4.4.3.2.2

Cancel the common factor.

$\frac{11 (2 \ln (11 x))}{11 \cdot 1} \frac{d}{d x} [x] + \ln^{2} (11 x) \frac{d}{d x} [x]$

Step 4.4.3.2.3

Rewrite the expression.

$\frac{2 \ln (11 x)}{1} \frac{d}{d x} [x] + \ln^{2} (11 x) \frac{d}{d x} [x]$

Step 4.4.3.2.4

Divide $2 \ln (11 x)$ by $1$ .

$2 \ln (11 x) \frac{d}{d x} [x] + \ln^{2} (11 x) \frac{d}{d x} [x]$

Step 4.5

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

$2 \ln (11 x) \cdot 1 + \ln^{2} (11 x) \frac{d}{d x} [x]$

Step 4.6

Multiply $2$ by $1$ .

$2 \ln (11 x) + \ln^{2} (11 x) \frac{d}{d x} [x]$

Step 4.7

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

$2 \ln (11 x) + \ln^{2} (11 x) \cdot 1$

Step 4.8

Simplify the expression.

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Step 4.8.1

Multiply $\ln^{2} (11 x)$ by $1$ .

$2 \ln (11 x) + \ln^{2} (11 x)$

Step 4.8.2

Reorder terms.

$\ln^{2} (11 x) + 2 \ln (11 x)$