Calculus Examples

$- \frac{3}{4} \cdot \log_{3} (16 p^{4}) - \frac{2}{3} \cdot \log_{3} (8 p^{3})$

Step 1

By the Sum Rule, the derivative of $- \frac{3}{4} \cdot \log_{3} (16 p^{4}) - \frac{2}{3} \cdot \log_{3} (8 p^{3})$ with respect to $p$ is $\frac{d}{d p} [- \frac{3}{4} \cdot \log_{3} (16 p^{4})] + \frac{d}{d p} [- \frac{2}{3} \cdot \log_{3} (8 p^{3})]$ .

$\frac{d}{d p} [- \frac{3}{4} \cdot \log_{3} (16 p^{4})] + \frac{d}{d p} [- \frac{2}{3} \cdot \log_{3} (8 p^{3})]$

Step 2

Evaluate $\frac{d}{d p} [- \frac{3}{4} \cdot \log_{3} (16 p^{4})]$ .

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Step 2.1

Since $- \frac{3}{4}$ is constant with respect to $p$ , the derivative of $- \frac{3}{4} \cdot \log_{3} (16 p^{4})$ with respect to $p$ is $- \frac{3}{4} \frac{d}{d p} [\log_{3} (16 p^{4})]$ .

$- \frac{3}{4} \frac{d}{d p} [\log_{3} (16 p^{4})] + \frac{d}{d p} [- \frac{2}{3} \cdot \log_{3} (8 p^{3})]$

Step 2.2

Differentiate using the chain rule, which states that $\frac{d}{d p} [f (g (p))]$ is $f' (g (p)) g' (p)$ where $f (p) = \log_{3} (p)$ and $g (p) = 16 p^{4}$ .

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Step 2.2.1

To apply the Chain Rule, set $u_{1}$ as $16 p^{4}$ .

$- \frac{3}{4} (\frac{d}{d u_{1}} [\log_{3} (u_{1})] \frac{d}{d p} [16 p^{4}]) + \frac{d}{d p} [- \frac{2}{3} \cdot \log_{3} (8 p^{3})]$

Step 2.2.2

The derivative of $\log_{3} (u_{1})$ with respect to $u_{1}$ is $\frac{1}{u_{1} \ln (3)}$ .

$- \frac{3}{4} (\frac{1}{u_{1} \ln (3)} \frac{d}{d p} [16 p^{4}]) + \frac{d}{d p} [- \frac{2}{3} \cdot \log_{3} (8 p^{3})]$

Step 2.2.3

Replace all occurrences of $u_{1}$ with $16 p^{4}$ .

$- \frac{3}{4} (\frac{1}{16 p^{4} \ln (3)} \frac{d}{d p} [16 p^{4}]) + \frac{d}{d p} [- \frac{2}{3} \cdot \log_{3} (8 p^{3})]$

Step 2.3

Since $16$ is constant with respect to $p$ , the derivative of $16 p^{4}$ with respect to $p$ is $16 \frac{d}{d p} [p^{4}]$ .

$- \frac{3}{4} (\frac{1}{16 p^{4} \ln (3)} (16 \frac{d}{d p} [p^{4}])) + \frac{d}{d p} [- \frac{2}{3} \cdot \log_{3} (8 p^{3})]$

Step 2.4

Differentiate using the Power Rule which states that $\frac{d}{d p} [p^{n}]$ is $n p^{n - 1}$ where $n = 4$ .

$- \frac{3}{4} (\frac{1}{16 p^{4} \ln (3)} (16 (4 p^{3}))) + \frac{d}{d p} [- \frac{2}{3} \cdot \log_{3} (8 p^{3})]$

Step 2.5

Multiply $4$ by $16$ .

$- \frac{3}{4} (\frac{1}{16 p^{4} \ln (3)} (64 p^{3})) + \frac{d}{d p} [- \frac{2}{3} \cdot \log_{3} (8 p^{3})]$

Step 2.6

Combine $64$ and $\frac{1}{16 p^{4} \ln (3)}$ .

$- \frac{3}{4} (\frac{64}{16 p^{4} \ln (3)} p^{3}) + \frac{d}{d p} [- \frac{2}{3} \cdot \log_{3} (8 p^{3})]$

Step 2.7

Combine $\frac{64}{16 p^{4} \ln (3)}$ and $p^{3}$ .

$- \frac{3}{4} \cdot \frac{64 p^{3}}{16 p^{4} \ln (3)} + \frac{d}{d p} [- \frac{2}{3} \cdot \log_{3} (8 p^{3})]$

Step 2.8

Cancel the common factor of $64$ and $16$ .

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Step 2.8.1

Factor $16$ out of $64 p^{3}$ .

$- \frac{3}{4} \cdot \frac{16 (4 p^{3})}{16 p^{4} \ln (3)} + \frac{d}{d p} [- \frac{2}{3} \cdot \log_{3} (8 p^{3})]$

Step 2.8.2

Cancel the common factors.

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Step 2.8.2.1

Factor $16$ out of $16 p^{4} \ln (3)$ .

$- \frac{3}{4} \cdot \frac{16 (4 p^{3})}{16 (p^{4} \ln (3))} + \frac{d}{d p} [- \frac{2}{3} \cdot \log_{3} (8 p^{3})]$

Step 2.8.2.2

Cancel the common factor.

$- \frac{3}{4} \cdot \frac{16 (4 p^{3})}{16 (p^{4} \ln (3))} + \frac{d}{d p} [- \frac{2}{3} \cdot \log_{3} (8 p^{3})]$

Step 2.8.2.3

Rewrite the expression.

$- \frac{3}{4} \cdot \frac{4 p^{3}}{p^{4} \ln (3)} + \frac{d}{d p} [- \frac{2}{3} \cdot \log_{3} (8 p^{3})]$

Step 2.9

Cancel the common factor of $p^{3}$ and $p^{4}$ .

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Step 2.9.1

Factor $p^{3}$ out of $4 p^{3}$ .

$- \frac{3}{4} \cdot \frac{p^{3} \cdot 4}{p^{4} \ln (3)} + \frac{d}{d p} [- \frac{2}{3} \cdot \log_{3} (8 p^{3})]$

Step 2.9.2

Cancel the common factors.

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Step 2.9.2.1

Factor $p^{3}$ out of $p^{4} \ln (3)$ .

$- \frac{3}{4} \cdot \frac{p^{3} \cdot 4}{p^{3} (p \ln (3))} + \frac{d}{d p} [- \frac{2}{3} \cdot \log_{3} (8 p^{3})]$

Step 2.9.2.2

Cancel the common factor.

$- \frac{3}{4} \cdot \frac{p^{3} \cdot 4}{p^{3} (p \ln (3))} + \frac{d}{d p} [- \frac{2}{3} \cdot \log_{3} (8 p^{3})]$

Step 2.9.2.3

Rewrite the expression.

$- \frac{3}{4} \cdot \frac{4}{p \ln (3)} + \frac{d}{d p} [- \frac{2}{3} \cdot \log_{3} (8 p^{3})]$

Step 2.10

Multiply $\frac{4}{p \ln (3)}$ by $\frac{3}{4}$ .

$- \frac{4 \cdot 3}{p \ln (3) \cdot 4} + \frac{d}{d p} [- \frac{2}{3} \cdot \log_{3} (8 p^{3})]$

Step 2.11

Multiply $4$ by $3$ .

$- \frac{12}{p \ln (3) \cdot 4} + \frac{d}{d p} [- \frac{2}{3} \cdot \log_{3} (8 p^{3})]$

Step 2.12

Move $4$ to the left of $p \ln (3)$ .

$- \frac{12}{4 \cdot (p \ln (3))} + \frac{d}{d p} [- \frac{2}{3} \cdot \log_{3} (8 p^{3})]$

Step 2.13

Cancel the common factor of $12$ and $4$ .

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Step 2.13.1

Factor $4$ out of $12$ .

$- \frac{4 \cdot 3}{4 p \ln (3)} + \frac{d}{d p} [- \frac{2}{3} \cdot \log_{3} (8 p^{3})]$

Step 2.13.2

Cancel the common factors.

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Step 2.13.2.1

Factor $4$ out of $4 p \ln (3)$ .

$- \frac{4 \cdot 3}{4 (p \ln (3))} + \frac{d}{d p} [- \frac{2}{3} \cdot \log_{3} (8 p^{3})]$

Step 2.13.2.2

Cancel the common factor.

$- \frac{4 \cdot 3}{4 (p \ln (3))} + \frac{d}{d p} [- \frac{2}{3} \cdot \log_{3} (8 p^{3})]$

Step 2.13.2.3

Rewrite the expression.

$- \frac{3}{p \ln (3)} + \frac{d}{d p} [- \frac{2}{3} \cdot \log_{3} (8 p^{3})]$

Step 3

Evaluate $\frac{d}{d p} [- \frac{2}{3} \cdot \log_{3} (8 p^{3})]$ .

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Step 3.1

Since $- \frac{2}{3}$ is constant with respect to $p$ , the derivative of $- \frac{2}{3} \cdot \log_{3} (8 p^{3})$ with respect to $p$ is $- \frac{2}{3} \frac{d}{d p} [\log_{3} (8 p^{3})]$ .

$- \frac{3}{p \ln (3)} - \frac{2}{3} \frac{d}{d p} [\log_{3} (8 p^{3})]$

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d p} [f (g (p))]$ is $f' (g (p)) g' (p)$ where $f (p) = \log_{3} (p)$ and $g (p) = 8 p^{3}$ .

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Step 3.2.1

To apply the Chain Rule, set $u_{2}$ as $8 p^{3}$ .

$- \frac{3}{p \ln (3)} - \frac{2}{3} (\frac{d}{d u_{2}} [\log_{3} (u_{2})] \frac{d}{d p} [8 p^{3}])$

Step 3.2.2

The derivative of $\log_{3} (u_{2})$ with respect to $u_{2}$ is $\frac{1}{u_{2} \ln (3)}$ .

$- \frac{3}{p \ln (3)} - \frac{2}{3} (\frac{1}{u_{2} \ln (3)} \frac{d}{d p} [8 p^{3}])$

Step 3.2.3

Replace all occurrences of $u_{2}$ with $8 p^{3}$ .

$- \frac{3}{p \ln (3)} - \frac{2}{3} (\frac{1}{8 p^{3} \ln (3)} \frac{d}{d p} [8 p^{3}])$

Step 3.3

Since $8$ is constant with respect to $p$ , the derivative of $8 p^{3}$ with respect to $p$ is $8 \frac{d}{d p} [p^{3}]$ .

$- \frac{3}{p \ln (3)} - \frac{2}{3} (\frac{1}{8 p^{3} \ln (3)} (8 \frac{d}{d p} [p^{3}]))$

Step 3.4

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

$- \frac{3}{p \ln (3)} - \frac{2}{3} (\frac{1}{8 p^{3} \ln (3)} (8 (3 p^{2})))$

Step 3.5

Multiply $3$ by $8$ .

$- \frac{3}{p \ln (3)} - \frac{2}{3} (\frac{1}{8 p^{3} \ln (3)} (24 p^{2}))$

Step 3.6

Combine $24$ and $\frac{1}{8 p^{3} \ln (3)}$ .

$- \frac{3}{p \ln (3)} - \frac{2}{3} (\frac{24}{8 p^{3} \ln (3)} p^{2})$

Step 3.7

Combine $\frac{24}{8 p^{3} \ln (3)}$ and $p^{2}$ .

$- \frac{3}{p \ln (3)} - \frac{2}{3} \cdot \frac{24 p^{2}}{8 p^{3} \ln (3)}$

Step 3.8

Cancel the common factor of $24$ and $8$ .

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Step 3.8.1

Factor $8$ out of $24 p^{2}$ .

$- \frac{3}{p \ln (3)} - \frac{2}{3} \cdot \frac{8 (3 p^{2})}{8 p^{3} \ln (3)}$

Step 3.8.2

Cancel the common factors.

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Step 3.8.2.1

Factor $8$ out of $8 p^{3} \ln (3)$ .

$- \frac{3}{p \ln (3)} - \frac{2}{3} \cdot \frac{8 (3 p^{2})}{8 (p^{3} \ln (3))}$

Step 3.8.2.2

Cancel the common factor.

$- \frac{3}{p \ln (3)} - \frac{2}{3} \cdot \frac{8 (3 p^{2})}{8 (p^{3} \ln (3))}$

Step 3.8.2.3

Rewrite the expression.

$- \frac{3}{p \ln (3)} - \frac{2}{3} \cdot \frac{3 p^{2}}{p^{3} \ln (3)}$

Step 3.9

Cancel the common factor of $p^{2}$ and $p^{3}$ .

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Step 3.9.1

Factor $p^{2}$ out of $3 p^{2}$ .

$- \frac{3}{p \ln (3)} - \frac{2}{3} \cdot \frac{p^{2} \cdot 3}{p^{3} \ln (3)}$

Step 3.9.2

Cancel the common factors.

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Step 3.9.2.1

Factor $p^{2}$ out of $p^{3} \ln (3)$ .

$- \frac{3}{p \ln (3)} - \frac{2}{3} \cdot \frac{p^{2} \cdot 3}{p^{2} (p \ln (3))}$

Step 3.9.2.2

Cancel the common factor.

$- \frac{3}{p \ln (3)} - \frac{2}{3} \cdot \frac{p^{2} \cdot 3}{p^{2} (p \ln (3))}$

Step 3.9.2.3

Rewrite the expression.

$- \frac{3}{p \ln (3)} - \frac{2}{3} \cdot \frac{3}{p \ln (3)}$

Step 3.10

Multiply $\frac{3}{p \ln (3)}$ by $\frac{2}{3}$ .

$- \frac{3}{p \ln (3)} - \frac{3 \cdot 2}{p \ln (3) \cdot 3}$

Step 3.11

Multiply $3$ by $2$ .

$- \frac{3}{p \ln (3)} - \frac{6}{p \ln (3) \cdot 3}$

Step 3.12

Move $3$ to the left of $p \ln (3)$ .

$- \frac{3}{p \ln (3)} - \frac{6}{3 \cdot (p \ln (3))}$

Step 3.13

Cancel the common factor of $6$ and $3$ .

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Step 3.13.1

Factor $3$ out of $6$ .

$- \frac{3}{p \ln (3)} - \frac{3 \cdot 2}{3 p \ln (3)}$

Step 3.13.2

Cancel the common factors.

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Step 3.13.2.1

Factor $3$ out of $3 p \ln (3)$ .

$- \frac{3}{p \ln (3)} - \frac{3 \cdot 2}{3 (p \ln (3))}$

Step 3.13.2.2

Cancel the common factor.

$- \frac{3}{p \ln (3)} - \frac{3 \cdot 2}{3 (p \ln (3))}$

Step 3.13.2.3

Rewrite the expression.

$- \frac{3}{p \ln (3)} - \frac{2}{p \ln (3)}$

Step 4

Combine terms.

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Step 4.1

Combine the numerators over the common denominator.

$\frac{- 3 - 2}{p \ln (3)}$

Step 4.2

Subtract $2$ from $- 3$ .

$\frac{- 5}{p \ln (3)}$

Step 4.3

Move the negative in front of the fraction.

$- \frac{5}{p \ln (3)}$