Calculus Examples

$f (x) = 5 - \ln ({(x^{2} - 3 x + 3)}^{3})$ , $x = 2$

Step 1

Find the derivative.

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Step 1.1

Differentiate.

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Step 1.1.1

By the Sum Rule, the derivative of $5 - \ln ({(x^{2} - 3 x + 3)}^{3})$ with respect to $x$ is $\frac{d}{d x} [5] + \frac{d}{d x} [- \ln ({(x^{2} - 3 x + 3)}^{3})]$ .

$\frac{d}{d x} [5] + \frac{d}{d x} [- \ln ({(x^{2} - 3 x + 3)}^{3})]$

Step 1.1.2

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- \ln ({(x^{2} - 3 x + 3)}^{3})]$

Step 1.2

Evaluate $\frac{d}{d x} [- \ln ({(x^{2} - 3 x + 3)}^{3})]$ .

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Step 1.2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \ln ({(x^{2} - 3 x + 3)}^{3})$ with respect to $x$ is $- \frac{d}{d x} [\ln ({(x^{2} - 3 x + 3)}^{3})]$ .

$0 - \frac{d}{d x} [\ln ({(x^{2} - 3 x + 3)}^{3})]$

Step 1.2.2

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

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Step 1.2.2.1

To apply the Chain Rule, set $u_{1}$ as ${(x^{2} - 3 x + 3)}^{3}$ .

$0 - (\frac{d}{d u_{1}} [\ln (u_{1})] \frac{d}{d x} [{(x^{2} - 3 x + 3)}^{3}])$

Step 1.2.2.2

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

$0 - (\frac{1}{u_{1}} \frac{d}{d x} [{(x^{2} - 3 x + 3)}^{3}])$

Step 1.2.2.3

Replace all occurrences of $u_{1}$ with ${(x^{2} - 3 x + 3)}^{3}$ .

$0 - (\frac{1}{{(x^{2} - 3 x + 3)}^{3}} \frac{d}{d x} [{(x^{2} - 3 x + 3)}^{3}])$

Step 1.2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = x^{2} - 3 x + 3$ .

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Step 1.2.3.1

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

$0 - (\frac{1}{{(x^{2} - 3 x + 3)}^{3}} (\frac{d}{d u_{2}} [{u_{2}}^{3}] \frac{d}{d x} [x^{2} - 3 x + 3]))$

Step 1.2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 3$ .

$0 - (\frac{1}{{(x^{2} - 3 x + 3)}^{3}} (3 {u_{2}}^{2} \frac{d}{d x} [x^{2} - 3 x + 3]))$

Step 1.2.3.3

Replace all occurrences of $u_{2}$ with $x^{2} - 3 x + 3$ .

$0 - (\frac{1}{{(x^{2} - 3 x + 3)}^{3}} (3 {(x^{2} - 3 x + 3)}^{2} \frac{d}{d x} [x^{2} - 3 x + 3]))$

Step 1.2.4

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

$0 - (\frac{1}{{(x^{2} - 3 x + 3)}^{3}} (3 {(x^{2} - 3 x + 3)}^{2} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [3])))$

Step 1.2.5

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

$0 - (\frac{1}{{(x^{2} - 3 x + 3)}^{3}} (3 {(x^{2} - 3 x + 3)}^{2} (2 x + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [3])))$

Step 1.2.6

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

$0 - (\frac{1}{{(x^{2} - 3 x + 3)}^{3}} (3 {(x^{2} - 3 x + 3)}^{2} (2 x - 3 \frac{d}{d x} [x] + \frac{d}{d x} [3])))$

Step 1.2.7

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

$0 - (\frac{1}{{(x^{2} - 3 x + 3)}^{3}} (3 {(x^{2} - 3 x + 3)}^{2} (2 x - 3 \cdot 1 + \frac{d}{d x} [3])))$

Step 1.2.8

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$0 - (\frac{1}{{(x^{2} - 3 x + 3)}^{3}} (3 {(x^{2} - 3 x + 3)}^{2} (2 x - 3 \cdot 1 + 0)))$

Step 1.2.9

Multiply $- 3$ by $1$ .

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

Step 1.2.10

Add $2 x - 3$ and $0$ .

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

Step 1.2.11

Combine $3$ and $\frac{1}{{(x^{2} - 3 x + 3)}^{3}}$ .

$0 - (\frac{3}{{(x^{2} - 3 x + 3)}^{3}} ({(x^{2} - 3 x + 3)}^{2} (2 x - 3)))$

Step 1.2.12

Combine ${(x^{2} - 3 x + 3)}^{2}$ and $\frac{3}{{(x^{2} - 3 x + 3)}^{3}}$ .

$0 - (\frac{{(x^{2} - 3 x + 3)}^{2} \cdot 3}{{(x^{2} - 3 x + 3)}^{3}} (2 x - 3))$

Step 1.2.13

Move $3$ to the left of ${(x^{2} - 3 x + 3)}^{2}$ .

$0 - (\frac{3 \cdot {(x^{2} - 3 x + 3)}^{2}}{{(x^{2} - 3 x + 3)}^{3}} (2 x - 3))$

Step 1.2.14

Cancel the common factor of ${(x^{2} - 3 x + 3)}^{2}$ and ${(x^{2} - 3 x + 3)}^{3}$ .

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Step 1.2.14.1

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

$0 - (\frac{{(x^{2} - 3 x + 3)}^{2} \cdot 3}{{(x^{2} - 3 x + 3)}^{3}} (2 x - 3))$

Step 1.2.14.2

Cancel the common factors.

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Step 1.2.14.2.1

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

$0 - (\frac{{(x^{2} - 3 x + 3)}^{2} \cdot 3}{{(x^{2} - 3 x + 3)}^{2} (x^{2} - 3 x + 3)} (2 x - 3))$

Step 1.2.14.2.2

Cancel the common factor.

$0 - (\frac{{(x^{2} - 3 x + 3)}^{2} \cdot 3}{{(x^{2} - 3 x + 3)}^{2} (x^{2} - 3 x + 3)} (2 x - 3))$

Step 1.2.14.2.3

Rewrite the expression.

$0 - (\frac{3}{x^{2} - 3 x + 3} (2 x - 3))$

$0 - \frac{3}{x^{2} - 3 x + 3} (2 x - 3)$

Step 1.3

Simplify.

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Step 1.3.1

Subtract $\frac{3}{x^{2} - 3 x + 3} (2 x - 3)$ from $0$ .

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

Step 1.3.2

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

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

Step 1.3.3

Apply the distributive property.

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

Step 1.3.4

Multiply $2$ by $- 1$ .

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

Step 1.3.5

Multiply $- 1$ by $- 3$ .

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

Step 1.3.6

Multiply $- 2 x + 3$ by $\frac{3}{x^{2} - 3 x + 3}$ .

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

Step 1.3.7

Move $3$ to the left of $- 2 x + 3$ .

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

Step 1.3.8

Factor $- 1$ out of $- 2 x$ .

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

Step 1.3.9

Rewrite $3$ as $- 1 (- 3)$ .

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

Step 1.3.10

Factor $- 1$ out of $- (2 x) - 1 (- 3)$ .

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

Step 1.3.11

Rewrite $- (2 x - 3)$ as $- 1 (2 x - 3)$ .

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

Step 1.3.12

Move the negative in front of the fraction.

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

Step 2

Replace the variable $x$ with $2$ in the expression.

$f' (2) = - \frac{3 (2 (2) - 3)}{{(2)}^{2} - 3 \cdot 2 + 3}$

Step 3

Simplify the numerator.

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Step 3.1

Multiply $2$ by $2$ .

$f' (2) = - \frac{3 (4 - 3)}{2^{2} - 3 \cdot 2 + 3}$

Step 3.2

Subtract $3$ from $4$ .

$f' (2) = - \frac{3 \cdot 1}{2^{2} - 3 \cdot 2 + 3}$

Step 4

Simplify the denominator.

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Step 4.1

Raise $2$ to the power of $2$ .

$f' (2) = - \frac{3 \cdot 1}{4 - 3 \cdot 2 + 3}$

Step 4.2

Multiply $- 3$ by $2$ .

$f' (2) = - \frac{3 \cdot 1}{4 - 6 + 3}$

Step 4.3

Subtract $6$ from $4$ .

$f' (2) = - \frac{3 \cdot 1}{- 2 + 3}$

Step 4.4

Add $- 2$ and $3$ .

$f' (2) = - \frac{3 \cdot 1}{1}$

Step 5

Simplify the expression.

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Step 5.1

Multiply $3$ by $1$ .

$f' (2) = - \frac{3}{1}$

Step 5.2

Divide $3$ by $1$ .

$f' (2) = - 1 \cdot 3$

Step 5.3

Multiply $- 1$ by $3$ .

$f' (2) = - 3$