Calculus Examples

$y = \ln (7 x^{3} - x^{2})$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\ln (7 x^{3} - x^{2}))$

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 $f (x) = \ln (x)$ and $g (x) = 7 x^{3} - x^{2}$ .

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Step 3.1.1

To apply the Chain Rule, set $u$ as $7 x^{3} - x^{2}$ .

$\frac{d}{d u} [\ln (u)] \frac{d}{d x} [7 x^{3} - x^{2}]$

Step 3.1.2

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

$\frac{1}{u} \frac{d}{d x} [7 x^{3} - x^{2}]$

Step 3.1.3

Replace all occurrences of $u$ with $7 x^{3} - x^{2}$ .

$\frac{1}{7 x^{3} - x^{2}} \frac{d}{d x} [7 x^{3} - x^{2}]$

Step 3.2

Differentiate.

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Step 3.2.1

By the Sum Rule, the derivative of $7 x^{3} - x^{2}$ with respect to $x$ is $\frac{d}{d x} [7 x^{3}] + \frac{d}{d x} [- x^{2}]$ .

$\frac{1}{7 x^{3} - x^{2}} (\frac{d}{d x} [7 x^{3}] + \frac{d}{d x} [- x^{2}])$

Step 3.2.2

Since $7$ is constant with respect to $x$ , the derivative of $7 x^{3}$ with respect to $x$ is $7 \frac{d}{d x} [x^{3}]$ .

$\frac{1}{7 x^{3} - x^{2}} (7 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- x^{2}])$

Step 3.2.3

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

$\frac{1}{7 x^{3} - x^{2}} (7 (3 x^{2}) + \frac{d}{d x} [- x^{2}])$

Step 3.2.4

Multiply $3$ by $7$ .

$\frac{1}{7 x^{3} - x^{2}} (21 x^{2} + \frac{d}{d x} [- x^{2}])$

Step 3.2.5

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

$\frac{1}{7 x^{3} - x^{2}} (21 x^{2} - \frac{d}{d x} [x^{2}])$

Step 3.2.6

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

$\frac{1}{7 x^{3} - x^{2}} (21 x^{2} - (2 x))$

Step 3.2.7

Multiply $2$ by $- 1$ .

$\frac{1}{7 x^{3} - x^{2}} (21 x^{2} - 2 x)$

Step 3.3

Simplify.

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Step 3.3.1

Reorder the factors of $\frac{1}{7 x^{3} - x^{2}} (21 x^{2} - 2 x)$ .

$(21 x^{2} - 2 x) \frac{1}{7 x^{3} - x^{2}}$

Step 3.3.2

Factor $x^{2}$ out of $7 x^{3} - x^{2}$ .

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Step 3.3.2.1

Factor $x^{2}$ out of $7 x^{3}$ .

$(21 x^{2} - 2 x) \frac{1}{x^{2} (7 x) - x^{2}}$

Step 3.3.2.2

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

$(21 x^{2} - 2 x) \frac{1}{x^{2} (7 x) + x^{2} \cdot - 1}$

Step 3.3.2.3

Factor $x^{2}$ out of $x^{2} (7 x) + x^{2} \cdot - 1$ .

$(21 x^{2} - 2 x) \frac{1}{x^{2} (7 x - 1)}$

Step 3.3.3

Multiply $21 x^{2} - 2 x$ by $\frac{1}{x^{2} (7 x - 1)}$ .

$\frac{21 x^{2} - 2 x}{x^{2} (7 x - 1)}$

Step 3.3.4

Factor $x$ out of $21 x^{2} - 2 x$ .

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Step 3.3.4.1

Factor $x$ out of $21 x^{2}$ .

$\frac{x (21 x) - 2 x}{x^{2} (7 x - 1)}$

Step 3.3.4.2

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

$\frac{x (21 x) + x \cdot - 2}{x^{2} (7 x - 1)}$

Step 3.3.4.3

Factor $x$ out of $x (21 x) + x \cdot - 2$ .

$\frac{x (21 x - 2)}{x^{2} (7 x - 1)}$

Step 3.3.5

Cancel the common factors.

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Step 3.3.5.1

Factor $x$ out of $x^{2} (7 x - 1)$ .

$\frac{x (21 x - 2)}{x (x (7 x - 1))}$

Step 3.3.5.2

Cancel the common factor.

$\frac{x (21 x - 2)}{x (x (7 x - 1))}$

Step 3.3.5.3

Rewrite the expression.

$\frac{21 x - 2}{x (7 x - 1)}$

Step 4

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

$y' = \frac{21 x - 2}{x (7 x - 1)}$

Step 5

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

$\frac{d y}{d x} = \frac{21 x - 2}{x (7 x - 1)}$