Calculus Examples

$g (x) = \ln (2 x^{3} - c o x \cdot x)$

Step 1

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

$\frac{d}{d x} [\ln (2 x^{3} - c o (x^{1} x))]$

Step 2

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

$\frac{d}{d x} [\ln (2 x^{3} - c o (x^{1} x^{1}))]$

Step 3

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

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

Step 4

Add $1$ and $1$ .

$\frac{d}{d x} [\ln (2 x^{3} - c o x^{2})]$

Step 5

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 $g (x) = 2 x^{3} - c o x^{2}$ .

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Step 5.1

To apply the Chain Rule, set $u$ as $2 x^{3} - c o x^{2}$ .

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

Step 5.2

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

$\frac{1}{u} \frac{d}{d x} [2 x^{3} - c o x^{2}]$

Step 5.3

Replace all occurrences of $u$ with $2 x^{3} - c o x^{2}$ .

$\frac{1}{2 x^{3} - c o x^{2}} \frac{d}{d x} [2 x^{3} - c o x^{2}]$

Step 6

Differentiate.

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Step 6.1

By the Sum Rule, the derivative of $2 x^{3} - c o x^{2}$ with respect to $x$ is $\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [- c o x^{2}]$ .

$\frac{1}{2 x^{3} - c o x^{2}} (\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [- c o x^{2}])$

Step 6.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{3}$ with respect to $x$ is $2 \frac{d}{d x} [x^{3}]$ .

$\frac{1}{2 x^{3} - c o x^{2}} (2 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- c o x^{2}])$

Step 6.3

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

$\frac{1}{2 x^{3} - c o x^{2}} (2 (3 x^{2}) + \frac{d}{d x} [- c o x^{2}])$

Step 6.4

Multiply $3$ by $2$ .

$\frac{1}{2 x^{3} - c o x^{2}} (6 x^{2} + \frac{d}{d x} [- c o x^{2}])$

Step 6.5

Since $- c o$ is constant with respect to $x$ , the derivative of $- c o x^{2}$ with respect to $x$ is $- c o \frac{d}{d x} [x^{2}]$ .

$\frac{1}{2 x^{3} - c o x^{2}} (6 x^{2} - c o \frac{d}{d x} [x^{2}])$

Step 6.6

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

$\frac{1}{2 x^{3} - c o x^{2}} (6 x^{2} - c o (2 x))$

Step 6.7

Multiply $2$ by $- 1$ .

$\frac{1}{2 x^{3} - c o x^{2}} (6 x^{2} - 2 c o x)$

Step 7

Simplify.

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Step 7.1

Reorder the factors of $\frac{1}{2 x^{3} - c o x^{2}} (6 x^{2} - 2 c o x)$ .

$(6 x^{2} - 2 c o x) \frac{1}{2 x^{3} - c o x^{2}}$

Step 7.2

Factor $x^{2}$ out of $2 x^{3} - c o x^{2}$ .

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Step 7.2.1

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

$(6 x^{2} - 2 c o x) \frac{1}{x^{2} (2 x) - c o x^{2}}$

Step 7.2.2

Factor $x^{2}$ out of $- c o x^{2}$ .

$(6 x^{2} - 2 c o x) \frac{1}{x^{2} (2 x) + x^{2} (- c o)}$

Step 7.2.3

Factor $x^{2}$ out of $x^{2} (2 x) + x^{2} (- c o)$ .

$(6 x^{2} - 2 c o x) \frac{1}{x^{2} (2 x - c o)}$

Step 7.3

Multiply $6 x^{2} - 2 c o x$ by $\frac{1}{x^{2} (2 x - c o)}$ .

$\frac{6 x^{2} - 2 c o x}{x^{2} (2 x - c o)}$

Step 7.4

Factor $2 x$ out of $6 x^{2} - 2 c o x$ .

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Step 7.4.1

Factor $2 x$ out of $6 x^{2}$ .

$\frac{2 x (3 x) - 2 c o x}{x^{2} (2 x - c o)}$

Step 7.4.2

Factor $2 x$ out of $- 2 c o x$ .

$\frac{2 x (3 x) + 2 x (- c o)}{x^{2} (2 x - c o)}$

Step 7.4.3

Factor $2 x$ out of $2 x (3 x) + 2 x (- c o)$ .

$\frac{2 x (3 x - c o)}{x^{2} (2 x - c o)}$

Step 7.5

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

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Step 7.5.1

Factor $x$ out of $2 x (3 x - c o)$ .

$\frac{x (2 (3 x - c o))}{x^{2} (2 x - c o)}$

Step 7.5.2

Cancel the common factors.

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Step 7.5.2.1

Factor $x$ out of $x^{2} (2 x - c o)$ .

$\frac{x (2 (3 x - c o))}{x (x (2 x - c o))}$

Step 7.5.2.2

Cancel the common factor.

$\frac{x (2 (3 x - c o))}{x (x (2 x - c o))}$

Step 7.5.2.3

Rewrite the expression.

$\frac{2 (3 x - c o)}{x (2 x - c o)}$