Calculus Examples

$\ln (- x^{3} + 6 x^{2} + 135 x + 1)$

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 $g (x) = - x^{3} + 6 x^{2} + 135 x + 1$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $- x^{3} + 6 x^{2} + 135 x + 1$ .

$\frac{d}{d u} [\ln (u)] \frac{d}{d x} [- x^{3} + 6 x^{2} + 135 x + 1]$

Step 1.2

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

$\frac{1}{u} \frac{d}{d x} [- x^{3} + 6 x^{2} + 135 x + 1]$

Step 1.3

Replace all occurrences of $u$ with $- x^{3} + 6 x^{2} + 135 x + 1$ .

$\frac{1}{- x^{3} + 6 x^{2} + 135 x + 1} \frac{d}{d x} [- x^{3} + 6 x^{2} + 135 x + 1]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $- x^{3} + 6 x^{2} + 135 x + 1$ with respect to $x$ is $\frac{d}{d x} [- x^{3}] + \frac{d}{d x} [6 x^{2}] + \frac{d}{d x} [135 x] + \frac{d}{d x} [1]$ .

$\frac{1}{- x^{3} + 6 x^{2} + 135 x + 1} (\frac{d}{d x} [- x^{3}] + \frac{d}{d x} [6 x^{2}] + \frac{d}{d x} [135 x] + \frac{d}{d x} [1])$

Step 2.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{3}$ with respect to $x$ is $- \frac{d}{d x} [x^{3}]$ .

$\frac{1}{- x^{3} + 6 x^{2} + 135 x + 1} (- \frac{d}{d x} [x^{3}] + \frac{d}{d x} [6 x^{2}] + \frac{d}{d x} [135 x] + \frac{d}{d x} [1])$

Step 2.3

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

$\frac{1}{- x^{3} + 6 x^{2} + 135 x + 1} (- (3 x^{2}) + \frac{d}{d x} [6 x^{2}] + \frac{d}{d x} [135 x] + \frac{d}{d x} [1])$

Step 2.4

Multiply $3$ by $- 1$ .

$\frac{1}{- x^{3} + 6 x^{2} + 135 x + 1} (- 3 x^{2} + \frac{d}{d x} [6 x^{2}] + \frac{d}{d x} [135 x] + \frac{d}{d x} [1])$

Step 2.5

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{2}$ with respect to $x$ is $6 \frac{d}{d x} [x^{2}]$ .

$\frac{1}{- x^{3} + 6 x^{2} + 135 x + 1} (- 3 x^{2} + 6 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [135 x] + \frac{d}{d x} [1])$

Step 2.6

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

$\frac{1}{- x^{3} + 6 x^{2} + 135 x + 1} (- 3 x^{2} + 6 (2 x) + \frac{d}{d x} [135 x] + \frac{d}{d x} [1])$

Step 2.7

Multiply $2$ by $6$ .

$\frac{1}{- x^{3} + 6 x^{2} + 135 x + 1} (- 3 x^{2} + 12 x + \frac{d}{d x} [135 x] + \frac{d}{d x} [1])$

Step 2.8

Since $135$ is constant with respect to $x$ , the derivative of $135 x$ with respect to $x$ is $135 \frac{d}{d x} [x]$ .

$\frac{1}{- x^{3} + 6 x^{2} + 135 x + 1} (- 3 x^{2} + 12 x + 135 \frac{d}{d x} [x] + \frac{d}{d x} [1])$

Step 2.9

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

$\frac{1}{- x^{3} + 6 x^{2} + 135 x + 1} (- 3 x^{2} + 12 x + 135 \cdot 1 + \frac{d}{d x} [1])$

Step 2.10

Multiply $135$ by $1$ .

$\frac{1}{- x^{3} + 6 x^{2} + 135 x + 1} (- 3 x^{2} + 12 x + 135 + \frac{d}{d x} [1])$

Step 2.11

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{1}{- x^{3} + 6 x^{2} + 135 x + 1} (- 3 x^{2} + 12 x + 135 + 0)$

Step 2.12

Add $- 3 x^{2} + 12 x + 135$ and $0$ .

$\frac{1}{- x^{3} + 6 x^{2} + 135 x + 1} (- 3 x^{2} + 12 x + 135)$

Step 3

Simplify.

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Step 3.1

Reorder the factors of $\frac{1}{- x^{3} + 6 x^{2} + 135 x + 1} (- 3 x^{2} + 12 x + 135)$ .

$(- 3 x^{2} + 12 x + 135) \frac{1}{- x^{3} + 6 x^{2} + 135 x + 1}$

Step 3.2

Multiply $- 3 x^{2} + 12 x + 135$ by $\frac{1}{- x^{3} + 6 x^{2} + 135 x + 1}$ .

$\frac{- 3 x^{2} + 12 x + 135}{- x^{3} + 6 x^{2} + 135 x + 1}$

Step 3.3

Simplify the numerator.

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Step 3.3.1

Factor $3$ out of $- 3 x^{2} + 12 x + 135$ .

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Step 3.3.1.1

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

$\frac{3 (- x^{2}) + 12 x + 135}{- x^{3} + 6 x^{2} + 135 x + 1}$

Step 3.3.1.2

Factor $3$ out of $12 x$ .

$\frac{3 (- x^{2}) + 3 (4 x) + 135}{- x^{3} + 6 x^{2} + 135 x + 1}$

Step 3.3.1.3

Factor $3$ out of $135$ .

$\frac{3 (- x^{2}) + 3 (4 x) + 3 (45)}{- x^{3} + 6 x^{2} + 135 x + 1}$

Step 3.3.1.4

Factor $3$ out of $3 (- x^{2}) + 3 (4 x)$ .

$\frac{3 (- x^{2} + 4 x) + 3 (45)}{- x^{3} + 6 x^{2} + 135 x + 1}$

Step 3.3.1.5

Factor $3$ out of $3 (- x^{2} + 4 x) + 3 (45)$ .

$\frac{3 (- x^{2} + 4 x + 45)}{- x^{3} + 6 x^{2} + 135 x + 1}$

Step 3.3.2

Factor by grouping.

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Step 3.3.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 45 = - 45$ and whose sum is $b = 4$ .

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Step 3.3.2.1.1

Factor $4$ out of $4 x$ .

$\frac{3 (- x^{2} + 4 (x) + 45)}{- x^{3} + 6 x^{2} + 135 x + 1}$

Step 3.3.2.1.2

Rewrite $4$ as $- 5$ plus $9$

$\frac{3 (- x^{2} + (- 5 + 9) x + 45)}{- x^{3} + 6 x^{2} + 135 x + 1}$

Step 3.3.2.1.3

Apply the distributive property.

$\frac{3 (- x^{2} - 5 x + 9 x + 45)}{- x^{3} + 6 x^{2} + 135 x + 1}$

Step 3.3.2.2

Factor out the greatest common factor from each group.

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Step 3.3.2.2.1

Group the first two terms and the last two terms.

$\frac{3 ((- x^{2} - 5 x) + 9 x + 45)}{- x^{3} + 6 x^{2} + 135 x + 1}$

Step 3.3.2.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{3 (x (- x - 5) - 9 (- x - 5))}{- x^{3} + 6 x^{2} + 135 x + 1}$

Step 3.3.2.3

Factor the polynomial by factoring out the greatest common factor, $- x - 5$ .

$\frac{3 ((- x - 5) (x - 9))}{- x^{3} + 6 x^{2} + 135 x + 1}$

$\frac{3 (- x - 5) (x - 9)}{- x^{3} + 6 x^{2} + 135 x + 1}$

Step 3.4

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

$\frac{3 (- (x) - 5) (x - 9)}{- x^{3} + 6 x^{2} + 135 x + 1}$

Step 3.5

Rewrite $- 5$ as $- 1 (5)$ .

$\frac{3 (- (x) - 1 (5)) (x - 9)}{- x^{3} + 6 x^{2} + 135 x + 1}$

Step 3.6

Factor $- 1$ out of $- (x) - 1 (5)$ .

$\frac{3 (- (x + 5)) (x - 9)}{- x^{3} + 6 x^{2} + 135 x + 1}$

Step 3.7

Rewrite $- (x + 5)$ as $- 1 (x + 5)$ .

$\frac{3 (- 1 (x + 5)) (x - 9)}{- x^{3} + 6 x^{2} + 135 x + 1}$

Step 3.8

Factor $- 1$ out of $- x^{3}$ .

$\frac{3 (- 1 (x + 5)) (x - 9)}{- (x^{3}) + 6 x^{2} + 135 x + 1}$

Step 3.9

Factor $- 1$ out of $6 x^{2}$ .

$\frac{3 (- 1 (x + 5)) (x - 9)}{- (x^{3}) - (- 6 x^{2}) + 135 x + 1}$

Step 3.10

Factor $- 1$ out of $- (x^{3}) - (- 6 x^{2})$ .

$\frac{3 (- 1 (x + 5)) (x - 9)}{- (x^{3} - 6 x^{2}) + 135 x + 1}$

Step 3.11

Factor $- 1$ out of $135 x$ .

$\frac{3 (- 1 (x + 5)) (x - 9)}{- (x^{3} - 6 x^{2}) - (- 135 x) + 1}$

Step 3.12

Factor $- 1$ out of $- (x^{3} - 6 x^{2}) - (- 135 x)$ .

$\frac{3 (- 1 (x + 5)) (x - 9)}{- (x^{3} - 6 x^{2} - 135 x) + 1}$

Step 3.13

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

$\frac{3 (- 1 (x + 5)) (x - 9)}{- (x^{3} - 6 x^{2} - 135 x) - 1 (- 1)}$

Step 3.14

Factor $- 1$ out of $- (x^{3} - 6 x^{2} - 135 x) - 1 (- 1)$ .

$\frac{3 (- 1 (x + 5)) (x - 9)}{- (x^{3} - 6 x^{2} - 135 x - 1)}$

Step 3.15

Rewrite $- (x^{3} - 6 x^{2} - 135 x - 1)$ as $- 1 (x^{3} - 6 x^{2} - 135 x - 1)$ .

$\frac{3 (- 1 (x + 5)) (x - 9)}{- 1 (x^{3} - 6 x^{2} - 135 x - 1)}$

Step 3.16

Cancel the common factor.

$\frac{3 (- 1 (x + 5)) (x - 9)}{- 1 (x^{3} - 6 x^{2} - 135 x - 1)}$

Step 3.17

Rewrite the expression.

$\frac{3 (x + 5) (x - 9)}{x^{3} - 6 x^{2} - 135 x - 1}$