Calculus Examples

$y = \ln (- 42 x^{2} + 11 x)$

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 $g (x) = - 42 x^{2} + 11 x$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $- 42 x^{2} + 11 x$ .

$\frac{d}{d u} [\ln (u)] \frac{d}{d x} [- 42 x^{2} + 11 x]$

Step 1.2

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

$\frac{1}{u} \frac{d}{d x} [- 42 x^{2} + 11 x]$

Step 1.3

Replace all occurrences of $u$ with $- 42 x^{2} + 11 x$ .

$\frac{1}{- 42 x^{2} + 11 x} \frac{d}{d x} [- 42 x^{2} + 11 x]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $- 42 x^{2} + 11 x$ with respect to $x$ is $\frac{d}{d x} [- 42 x^{2}] + \frac{d}{d x} [11 x]$ .

$\frac{1}{- 42 x^{2} + 11 x} (\frac{d}{d x} [- 42 x^{2}] + \frac{d}{d x} [11 x])$

Step 2.2

Since $- 42$ is constant with respect to $x$ , the derivative of $- 42 x^{2}$ with respect to $x$ is $- 42 \frac{d}{d x} [x^{2}]$ .

$\frac{1}{- 42 x^{2} + 11 x} (- 42 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [11 x])$

Step 2.3

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

$\frac{1}{- 42 x^{2} + 11 x} (- 42 (2 x) + \frac{d}{d x} [11 x])$

Step 2.4

Multiply $2$ by $- 42$ .

$\frac{1}{- 42 x^{2} + 11 x} (- 84 x + \frac{d}{d x} [11 x])$

Step 2.5

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

$\frac{1}{- 42 x^{2} + 11 x} (- 84 x + 11 \frac{d}{d x} [x])$

Step 2.6

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

$\frac{1}{- 42 x^{2} + 11 x} (- 84 x + 11 \cdot 1)$

Step 2.7

Multiply $11$ by $1$ .

$\frac{1}{- 42 x^{2} + 11 x} (- 84 x + 11)$

Step 3

Simplify.

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Step 3.1

Reorder the factors of $\frac{1}{- 42 x^{2} + 11 x} (- 84 x + 11)$ .

$(- 84 x + 11) \frac{1}{- 42 x^{2} + 11 x}$

Step 3.2

Factor $x$ out of $- 42 x^{2} + 11 x$ .

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Step 3.2.1

Factor $x$ out of $- 42 x^{2}$ .

$(- 84 x + 11) \frac{1}{x (- 42 x) + 11 x}$

Step 3.2.2

Factor $x$ out of $11 x$ .

$(- 84 x + 11) \frac{1}{x (- 42 x) + x \cdot 11}$

Step 3.2.3

Factor $x$ out of $x (- 42 x) + x \cdot 11$ .

$(- 84 x + 11) \frac{1}{x (- 42 x + 11)}$

Step 3.3

Multiply $- 84 x + 11$ by $\frac{1}{x (- 42 x + 11)}$ .

$\frac{- 84 x + 11}{x (- 42 x + 11)}$

Step 3.4

Factor $- 1$ out of $- 84 x$ .

$\frac{- (84 x) + 11}{x (- 42 x + 11)}$

Step 3.5

Rewrite $11$ as $- 1 (- 11)$ .

$\frac{- (84 x) - 1 (- 11)}{x (- 42 x + 11)}$

Step 3.6

Factor $- 1$ out of $- (84 x) - 1 (- 11)$ .

$\frac{- (84 x - 11)}{x (- 42 x + 11)}$

Step 3.7

Rewrite $- (84 x - 11)$ as $- 1 (84 x - 11)$ .

$\frac{- 1 (84 x - 11)}{x (- 42 x + 11)}$

Step 3.8

Factor $- 1$ out of $- 42 x$ .

$\frac{- 1 (84 x - 11)}{x (- (42 x) + 11)}$

Step 3.9

Rewrite $11$ as $- 1 (- 11)$ .

$\frac{- 1 (84 x - 11)}{x (- (42 x) - 1 (- 11))}$

Step 3.10

Factor $- 1$ out of $- (42 x) - 1 (- 11)$ .

$\frac{- 1 (84 x - 11)}{x (- (42 x - 11))}$

Step 3.11

Rewrite $- (42 x - 11)$ as $- 1 (42 x - 11)$ .

$\frac{- 1 (84 x - 11)}{x (- 1 (42 x - 11))}$

Step 3.12

Cancel the common factor.

$\frac{- 1 (84 x - 11)}{x (- 1 (42 x - 11))}$

Step 3.13

Rewrite the expression.

$\frac{84 x - 11}{x (42 x - 11)}$