Calculus Examples

$y = \log_{6} ({(x^{2} + 2 x)}^{\frac{7}{2}})$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \log_{6} (x)$ and $g (x) = {(x^{2} + 2 x)}^{\frac{7}{2}}$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as ${(x^{2} + 2 x)}^{\frac{7}{2}}$ .

$\frac{d}{d u_{1}} [\log_{6} (u_{1})] \frac{d}{d x} [{(x^{2} + 2 x)}^{\frac{7}{2}}]$

Step 1.2

The derivative of $\log_{6} (u_{1})$ with respect to $u_{1}$ is $\frac{1}{u_{1} \ln (6)}$ .

$\frac{1}{u_{1} \ln (6)} \frac{d}{d x} [{(x^{2} + 2 x)}^{\frac{7}{2}}]$

Step 1.3

Replace all occurrences of $u_{1}$ with ${(x^{2} + 2 x)}^{\frac{7}{2}}$ .

$\frac{1}{{(x^{2} + 2 x)}^{\frac{7}{2}} \ln (6)} \frac{d}{d x} [{(x^{2} + 2 x)}^{\frac{7}{2}}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{7}{2}}$ and $g (x) = x^{2} + 2 x$ .

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Step 2.1

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

$\frac{1}{{(x^{2} + 2 x)}^{\frac{7}{2}} \ln (6)} (\frac{d}{d u_{2}} [{u_{2}}^{\frac{7}{2}}] \frac{d}{d x} [x^{2} + 2 x])$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = \frac{7}{2}$ .

$\frac{1}{{(x^{2} + 2 x)}^{\frac{7}{2}} \ln (6)} (\frac{7}{2} {u_{2}}^{\frac{7}{2} - 1} \frac{d}{d x} [x^{2} + 2 x])$

Step 2.3

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

$\frac{1}{{(x^{2} + 2 x)}^{\frac{7}{2}} \ln (6)} (\frac{7}{2} {(x^{2} + 2 x)}^{\frac{7}{2} - 1} \frac{d}{d x} [x^{2} + 2 x])$

Step 3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{{(x^{2} + 2 x)}^{\frac{7}{2}} \ln (6)} (\frac{7}{2} {(x^{2} + 2 x)}^{\frac{7}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [x^{2} + 2 x])$

Step 4

Combine $- 1$ and $\frac{2}{2}$ .

$\frac{1}{{(x^{2} + 2 x)}^{\frac{7}{2}} \ln (6)} (\frac{7}{2} {(x^{2} + 2 x)}^{\frac{7}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [x^{2} + 2 x])$

Step 5

Combine the numerators over the common denominator.

$\frac{1}{{(x^{2} + 2 x)}^{\frac{7}{2}} \ln (6)} (\frac{7}{2} {(x^{2} + 2 x)}^{\frac{7 - 1 \cdot 2}{2}} \frac{d}{d x} [x^{2} + 2 x])$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

$\frac{1}{{(x^{2} + 2 x)}^{\frac{7}{2}} \ln (6)} (\frac{7}{2} {(x^{2} + 2 x)}^{\frac{7 - 2}{2}} \frac{d}{d x} [x^{2} + 2 x])$

Step 6.2

Subtract $2$ from $7$ .

$\frac{1}{{(x^{2} + 2 x)}^{\frac{7}{2}} \ln (6)} (\frac{7}{2} {(x^{2} + 2 x)}^{\frac{5}{2}} \frac{d}{d x} [x^{2} + 2 x])$

Step 7

Combine $\frac{7}{2}$ and ${(x^{2} + 2 x)}^{\frac{5}{2}}$ .

$\frac{1}{{(x^{2} + 2 x)}^{\frac{7}{2}} \ln (6)} (\frac{7 {(x^{2} + 2 x)}^{\frac{5}{2}}}{2} \frac{d}{d x} [x^{2} + 2 x])$

Step 8

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

$\frac{1}{{(x^{2} + 2 x)}^{\frac{7}{2}} \ln (6)} (\frac{7 {(x^{2} + 2 x)}^{\frac{5}{2}}}{2} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 x]))$

Step 9

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

$\frac{1}{{(x^{2} + 2 x)}^{\frac{7}{2}} \ln (6)} (\frac{7 {(x^{2} + 2 x)}^{\frac{5}{2}}}{2} (2 x + \frac{d}{d x} [2 x]))$

Step 10

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

$\frac{1}{{(x^{2} + 2 x)}^{\frac{7}{2}} \ln (6)} (\frac{7 {(x^{2} + 2 x)}^{\frac{5}{2}}}{2} (2 x + 2 \frac{d}{d x} [x]))$

Step 11

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

$\frac{1}{{(x^{2} + 2 x)}^{\frac{7}{2}} \ln (6)} (\frac{7 {(x^{2} + 2 x)}^{\frac{5}{2}}}{2} (2 x + 2 \cdot 1))$

Step 12

Simplify the expression.

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Step 12.1

Multiply $2$ by $1$ .

$\frac{1}{{(x^{2} + 2 x)}^{\frac{7}{2}} \ln (6)} (\frac{7 {(x^{2} + 2 x)}^{\frac{5}{2}}}{2} (2 x + 2))$

Step 12.2

Reorder the factors of $\frac{7 {(x^{2} + 2 x)}^{\frac{5}{2}}}{2} (2 x + 2)$ .

$\frac{1}{{(x^{2} + 2 x)}^{\frac{7}{2}} \ln (6)} ((2 x + 2) \frac{7 {(x^{2} + 2 x)}^{\frac{5}{2}}}{2})$

Step 13

Simplify.

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Step 13.1

Combine terms.

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Step 13.1.1

Multiply $\frac{7 {(x^{2} + 2 x)}^{\frac{5}{2}}}{2}$ by $\frac{1}{{(x^{2} + 2 x)}^{\frac{7}{2}} \ln (6)}$ .

$\frac{7 {(x^{2} + 2 x)}^{\frac{5}{2}}}{2 ({(x^{2} + 2 x)}^{\frac{7}{2}} \ln (6))} (2 x + 2)$

Step 13.1.2

Move ${(x^{2} + 2 x)}^{\frac{5}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{7}{2 {(x^{2} + 2 x)}^{\frac{7}{2}} \ln (6) {(x^{2} + 2 x)}^{- \frac{5}{2}}} (2 x + 2)$

Step 13.1.3

Simplify the denominator.

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Step 13.1.3.1

Multiply ${(x^{2} + 2 x)}^{\frac{7}{2}}$ by ${(x^{2} + 2 x)}^{- \frac{5}{2}}$ by adding the exponents.

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Step 13.1.3.1.1

Move ${(x^{2} + 2 x)}^{- \frac{5}{2}}$ .

$\frac{7}{2 ({(x^{2} + 2 x)}^{- \frac{5}{2}} {(x^{2} + 2 x)}^{\frac{7}{2}}) \ln (6)} (2 x + 2)$

Step 13.1.3.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{7}{2 {(x^{2} + 2 x)}^{- \frac{5}{2} + \frac{7}{2}} \ln (6)} (2 x + 2)$

Step 13.1.3.1.3

Combine the numerators over the common denominator.

$\frac{7}{2 {(x^{2} + 2 x)}^{\frac{- 5 + 7}{2}} \ln (6)} (2 x + 2)$

Step 13.1.3.1.4

Add $- 5$ and $7$ .

$\frac{7}{2 {(x^{2} + 2 x)}^{\frac{2}{2}} \ln (6)} (2 x + 2)$

Step 13.1.3.1.5

Divide $2$ by $2$ .

$\frac{7}{2 {(x^{2} + 2 x)}^{1} \ln (6)} (2 x + 2)$

Step 13.1.3.2

Simplify $2 {(x^{2} + 2 x)}^{1} \ln (6)$ .

$\frac{7}{2 (x^{2} + 2 x) \ln (6)} (2 x + 2)$

Step 13.2

Reorder the factors of $\frac{7}{2 (x^{2} + 2 x) \ln (6)} (2 x + 2)$ .

$(2 x + 2) \frac{7}{2 (x^{2} + 2 x) \ln (6)}$

Step 13.3

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

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Step 13.3.1

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

$(2 x + 2) \frac{7}{2 (x \cdot x + 2 x) \ln (6)}$

Step 13.3.2

Factor $x$ out of $2 x$ .

$(2 x + 2) \frac{7}{2 (x \cdot x + x \cdot 2) \ln (6)}$

Step 13.3.3

Factor $x$ out of $x \cdot x + x \cdot 2$ .

$(2 x + 2) \frac{7}{2 (x (x + 2)) \ln (6)}$

$(2 x + 2) \frac{7}{2 x (x + 2) \ln (6)}$

Step 13.4

Multiply $2 x + 2$ by $\frac{7}{2 x (x + 2) \ln (6)}$ .

$\frac{(2 x + 2) \cdot 7}{2 x (x + 2) \ln (6)}$

Step 13.5

Cancel the common factor of $2 x + 2$ and $2$ .

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Step 13.5.1

Factor $2$ out of $(2 x + 2) \cdot 7$ .

$\frac{2 ((x + 1) \cdot 7)}{2 x (x + 2) \ln (6)}$

Step 13.5.2

Cancel the common factors.

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Step 13.5.2.1

Factor $2$ out of $2 x (x + 2) \ln (6)$ .

$\frac{2 ((x + 1) \cdot 7)}{2 ((x (x + 2)) \ln (6))}$

Step 13.5.2.2

Cancel the common factor.

$\frac{2 ((x + 1) \cdot 7)}{2 ((x (x + 2)) \ln (6))}$

Step 13.5.2.3

Rewrite the expression.

$\frac{(x + 1) \cdot 7}{(x (x + 2)) \ln (6)}$

Step 13.6

Move $7$ to the left of $x + 1$ .

$\frac{7 \cdot (x + 1)}{x (x + 2) \ln (6)}$

Step 13.7

Reorder factors in $\frac{7 (x + 1)}{x (x + 2) \ln (6)}$ .

$\frac{7 (x + 1)}{x \ln (6) (x + 2)}$