Calculus Examples

$y = \ln (9 x^{3} - x^{2})$

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 $g (x) = 9 x^{3} - x^{2}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $9 x^{3} - x^{2}$ .

$\frac{d}{d u} [\ln (u)] \frac{d}{d x} [9 x^{3} - x^{2}]$

Step 1.2

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

$\frac{1}{u} \frac{d}{d x} [9 x^{3} - x^{2}]$

Step 1.3

Replace all occurrences of $u$ with $9 x^{3} - x^{2}$ .

$\frac{1}{9 x^{3} - x^{2}} \frac{d}{d x} [9 x^{3} - x^{2}]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $9 x^{3} - x^{2}$ with respect to $x$ is $\frac{d}{d x} [9 x^{3}] + \frac{d}{d x} [- x^{2}]$ .

$\frac{1}{9 x^{3} - x^{2}} (\frac{d}{d x} [9 x^{3}] + \frac{d}{d x} [- x^{2}])$

Step 2.2

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{3}$ with respect to $x$ is $9 \frac{d}{d x} [x^{3}]$ .

$\frac{1}{9 x^{3} - x^{2}} (9 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- x^{2}])$

Step 2.3

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

$\frac{1}{9 x^{3} - x^{2}} (9 (3 x^{2}) + \frac{d}{d x} [- x^{2}])$

Step 2.4

Multiply $3$ by $9$ .

$\frac{1}{9 x^{3} - x^{2}} (27 x^{2} + \frac{d}{d x} [- x^{2}])$

Step 2.5

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$\frac{1}{9 x^{3} - x^{2}} (27 x^{2} - \frac{d}{d x} [x^{2}])$

Step 2.6

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

$\frac{1}{9 x^{3} - x^{2}} (27 x^{2} - (2 x))$

Step 2.7

Multiply $2$ by $- 1$ .

$\frac{1}{9 x^{3} - x^{2}} (27 x^{2} - 2 x)$

Step 3

Simplify.

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Step 3.1

Reorder the factors of $\frac{1}{9 x^{3} - x^{2}} (27 x^{2} - 2 x)$ .

$(27 x^{2} - 2 x) \frac{1}{9 x^{3} - x^{2}}$

Step 3.2

Factor $x^{2}$ out of $9 x^{3} - x^{2}$ .

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Step 3.2.1

Factor $x^{2}$ out of $9 x^{3}$ .

$(27 x^{2} - 2 x) \frac{1}{x^{2} (9 x) - x^{2}}$

Step 3.2.2

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

$(27 x^{2} - 2 x) \frac{1}{x^{2} (9 x) + x^{2} \cdot - 1}$

Step 3.2.3

Factor $x^{2}$ out of $x^{2} (9 x) + x^{2} \cdot - 1$ .

$(27 x^{2} - 2 x) \frac{1}{x^{2} (9 x - 1)}$

Step 3.3

Multiply $27 x^{2} - 2 x$ by $\frac{1}{x^{2} (9 x - 1)}$ .

$\frac{27 x^{2} - 2 x}{x^{2} (9 x - 1)}$

Step 3.4

Factor $x$ out of $27 x^{2} - 2 x$ .

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Step 3.4.1

Factor $x$ out of $27 x^{2}$ .

$\frac{x (27 x) - 2 x}{x^{2} (9 x - 1)}$

Step 3.4.2

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

$\frac{x (27 x) + x \cdot - 2}{x^{2} (9 x - 1)}$

Step 3.4.3

Factor $x$ out of $x (27 x) + x \cdot - 2$ .

$\frac{x (27 x - 2)}{x^{2} (9 x - 1)}$

Step 3.5

Cancel the common factors.

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Step 3.5.1

Factor $x$ out of $x^{2} (9 x - 1)$ .

$\frac{x (27 x - 2)}{x (x (9 x - 1))}$

Step 3.5.2

Cancel the common factor.

$\frac{x (27 x - 2)}{x (x (9 x - 1))}$

Step 3.5.3

Rewrite the expression.

$\frac{27 x - 2}{x (9 x - 1)}$