Calculus Examples

$(- 3 k) {(\frac{- 2 k^{- 5}}{k^{3}})}^{- 4}$

Step 1

Move $k^{- 5}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{d}{d k} [- 3 k {(\frac{- 2}{k^{3} k^{5}})}^{- 4}]$

Step 2

Multiply $k^{3}$ by $k^{5}$ by adding the exponents.

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Step 2.1

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

$\frac{d}{d k} [- 3 k {(\frac{- 2}{k^{3 + 5}})}^{- 4}]$

Step 2.2

Add $3$ and $5$ .

$\frac{d}{d k} [- 3 k {(\frac{- 2}{k^{8}})}^{- 4}]$

Step 3

Move the negative in front of the fraction.

$\frac{d}{d k} [- 3 k {(- \frac{2}{k^{8}})}^{- 4}]$

Step 4

Factor $- 1$ out of $- \frac{2}{k^{8}}$ .

$\frac{d}{d k} [- 3 k {(- (\frac{2}{k^{8}}))}^{- 4}]$

Step 5

Apply the product rule to $- (\frac{2}{k^{8}})$ .

$\frac{d}{d k} [- 3 k ({(- 1)}^{- 4} {(\frac{2}{k^{8}})}^{- 4})]$

Step 6

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{d}{d k} [- 3 k (\frac{1}{{(- 1)}^{4}} {(\frac{2}{k^{8}})}^{- 4})]$

Step 7

Differentiate using the Constant Multiple Rule.

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Step 7.1

Raise $- 1$ to the power of $4$ .

$\frac{d}{d k} [- 3 k (\frac{1}{1} {(\frac{2}{k^{8}})}^{- 4})]$

Step 7.2

Reduce the expression by cancelling the common factors.

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Step 7.2.1

Cancel the common factor of $1$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{d}{d k} [- 3 k (\frac{1}{1} {(\frac{2}{k^{8}})}^{- 4})]$

Step 7.2.1.2

Rewrite the expression.

$\frac{d}{d k} [- 3 k (1 {(\frac{2}{k^{8}})}^{- 4})]$

Step 7.2.2

Multiply ${(\frac{2}{k^{8}})}^{- 4}$ by $1$ .

$\frac{d}{d k} [- 3 k {(\frac{2}{k^{8}})}^{- 4}]$

Step 7.3

Since $- 3$ is constant with respect to $k$ , the derivative of $- 3 k {(\frac{2}{k^{8}})}^{- 4}$ with respect to $k$ is $- 3 \frac{d}{d k} [k {(\frac{2}{k^{8}})}^{- 4}]$ .

$- 3 \frac{d}{d k} [k {(\frac{2}{k^{8}})}^{- 4}]$

Step 8

Differentiate using the Product Rule which states that $\frac{d}{d k} [f (k) g (k)]$ is $f (k) \frac{d}{d k} [g (k)] + g (k) \frac{d}{d k} [f (k)]$ where $f (k) = k$ and $g (k) = {(\frac{2}{k^{8}})}^{- 4}$ .

$- 3 (k \frac{d}{d k} [{(\frac{2}{k^{8}})}^{- 4}] + {(\frac{2}{k^{8}})}^{- 4} \frac{d}{d k} [k])$

Step 9

Differentiate using the chain rule, which states that $\frac{d}{d k} [f (g (k))]$ is $f' (g (k)) g' (k)$ where $f (k) = k^{- 4}$ and $g (k) = \frac{2}{k^{8}}$ .

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Step 9.1

To apply the Chain Rule, set $u$ as $\frac{2}{k^{8}}$ .

$- 3 (k (\frac{d}{d u} [u^{- 4}] \frac{d}{d k} [\frac{2}{k^{8}}]) + {(\frac{2}{k^{8}})}^{- 4} \frac{d}{d k} [k])$

Step 9.2

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

$- 3 (k (- 4 u^{- 5} \frac{d}{d k} [\frac{2}{k^{8}}]) + {(\frac{2}{k^{8}})}^{- 4} \frac{d}{d k} [k])$

Step 9.3

Replace all occurrences of $u$ with $\frac{2}{k^{8}}$ .

$- 3 (k (- 4 {(\frac{2}{k^{8}})}^{- 5} \frac{d}{d k} [\frac{2}{k^{8}}]) + {(\frac{2}{k^{8}})}^{- 4} \frac{d}{d k} [k])$

Step 10

Differentiate.

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Step 10.1

Since $2$ is constant with respect to $k$ , the derivative of $\frac{2}{k^{8}}$ with respect to $k$ is $2 \frac{d}{d k} [\frac{1}{k^{8}}]$ .

$- 3 (k (- 4 {(\frac{2}{k^{8}})}^{- 5} (2 \frac{d}{d k} [\frac{1}{k^{8}}])) + {(\frac{2}{k^{8}})}^{- 4} \frac{d}{d k} [k])$

Step 10.2

Simplify the expression.

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Step 10.2.1

Multiply $2$ by $- 4$ .

$- 3 (k (- 8 {(\frac{2}{k^{8}})}^{- 5} \frac{d}{d k} [\frac{1}{k^{8}}]) + {(\frac{2}{k^{8}})}^{- 4} \frac{d}{d k} [k])$

Step 10.2.2

Rewrite $\frac{1}{k^{8}}$ as ${(k^{8})}^{- 1}$ .

$- 3 (k (- 8 {(\frac{2}{k^{8}})}^{- 5} \frac{d}{d k} [{(k^{8})}^{- 1}]) + {(\frac{2}{k^{8}})}^{- 4} \frac{d}{d k} [k])$

Step 10.2.3

Multiply the exponents in ${(k^{8})}^{- 1}$ .

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Step 10.2.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- 3 (k (- 8 {(\frac{2}{k^{8}})}^{- 5} \frac{d}{d k} [k^{8 \cdot - 1}]) + {(\frac{2}{k^{8}})}^{- 4} \frac{d}{d k} [k])$

Step 10.2.3.2

Multiply $8$ by $- 1$ .

$- 3 (k (- 8 {(\frac{2}{k^{8}})}^{- 5} \frac{d}{d k} [k^{- 8}]) + {(\frac{2}{k^{8}})}^{- 4} \frac{d}{d k} [k])$

Step 10.3

Differentiate using the Power Rule which states that $\frac{d}{d k} [k^{n}]$ is $n k^{n - 1}$ where $n = - 8$ .

$- 3 (k (- 8 {(\frac{2}{k^{8}})}^{- 5} (- 8 k^{- 9})) + {(\frac{2}{k^{8}})}^{- 4} \frac{d}{d k} [k])$

Step 10.4

Multiply $- 8$ by $- 8$ .

$- 3 (k (64 {(\frac{2}{k^{8}})}^{- 5} k^{- 9}) + {(\frac{2}{k^{8}})}^{- 4} \frac{d}{d k} [k])$

Step 11

Raise $k$ to the power of $1$ .

$- 3 (k^{- 9} k^{1} (64 {(\frac{2}{k^{8}})}^{- 5}) + {(\frac{2}{k^{8}})}^{- 4} \frac{d}{d k} [k])$

Step 12

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

$- 3 (k^{- 9 + 1} (64 {(\frac{2}{k^{8}})}^{- 5}) + {(\frac{2}{k^{8}})}^{- 4} \frac{d}{d k} [k])$

Step 13

Add $- 9$ and $1$ .

$- 3 (k^{- 8} (64 {(\frac{2}{k^{8}})}^{- 5}) + {(\frac{2}{k^{8}})}^{- 4} \frac{d}{d k} [k])$

Step 14

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

$- 3 (k^{- 8} (64 {(\frac{2}{k^{8}})}^{- 5}) + {(\frac{2}{k^{8}})}^{- 4} \cdot 1)$

Step 15

Multiply ${(\frac{2}{k^{8}})}^{- 4}$ by $1$ .

$- 3 (k^{- 8} (64 {(\frac{2}{k^{8}})}^{- 5}) + {(\frac{2}{k^{8}})}^{- 4})$

Step 16

Simplify.

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Step 16.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 3 (\frac{1}{k^{8}} (64 {(\frac{2}{k^{8}})}^{- 5}) + {(\frac{2}{k^{8}})}^{- 4})$

Step 16.2

Change the sign of the exponent by rewriting the base as its reciprocal.

$- 3 (\frac{1}{k^{8}} (64 {(\frac{k^{8}}{2})}^{5}) + {(\frac{2}{k^{8}})}^{- 4})$

Step 16.3

Change the sign of the exponent by rewriting the base as its reciprocal.

$- 3 (\frac{1}{k^{8}} (64 {(\frac{k^{8}}{2})}^{5}) + {(\frac{k^{8}}{2})}^{4})$

Step 16.4

Apply the product rule to $\frac{k^{8}}{2}$ .

$- 3 (\frac{1}{k^{8}} (64 \frac{{(k^{8})}^{5}}{2^{5}}) + {(\frac{k^{8}}{2})}^{4})$

Step 16.5

Apply the product rule to $\frac{k^{8}}{2}$ .

$- 3 (\frac{1}{k^{8}} (64 \frac{{(k^{8})}^{5}}{2^{5}}) + \frac{{(k^{8})}^{4}}{2^{4}})$

Step 16.6

Apply the distributive property.

$- 3 (\frac{1}{k^{8}} (64 \frac{{(k^{8})}^{5}}{2^{5}})) - 3 \frac{{(k^{8})}^{4}}{2^{4}}$

Step 16.7

Combine terms.

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Step 16.7.1

Multiply the exponents in ${(k^{8})}^{5}$ .

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Step 16.7.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- 3 (\frac{1}{k^{8}} (64 \frac{k^{8 \cdot 5}}{2^{5}})) - 3 \frac{{(k^{8})}^{4}}{2^{4}}$

Step 16.7.1.2

Multiply $8$ by $5$ .

$- 3 (\frac{1}{k^{8}} (64 \frac{k^{40}}{2^{5}})) - 3 \frac{{(k^{8})}^{4}}{2^{4}}$

Step 16.7.2

Raise $2$ to the power of $5$ .

$- 3 (\frac{1}{k^{8}} (64 \frac{k^{40}}{32})) - 3 \frac{{(k^{8})}^{4}}{2^{4}}$

Step 16.7.3

Combine $64$ and $\frac{k^{40}}{32}$ .

$- 3 (\frac{1}{k^{8}} \cdot \frac{64 k^{40}}{32}) - 3 \frac{{(k^{8})}^{4}}{2^{4}}$

Step 16.7.4

Cancel the common factor of $64$ and $32$ .

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Step 16.7.4.1

Factor $32$ out of $64 k^{40}$ .

$- 3 (\frac{1}{k^{8}} \cdot \frac{32 (2 k^{40})}{32}) - 3 \frac{{(k^{8})}^{4}}{2^{4}}$

Step 16.7.4.2

Cancel the common factors.

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Step 16.7.4.2.1

Factor $32$ out of $32$ .

$- 3 (\frac{1}{k^{8}} \cdot \frac{32 (2 k^{40})}{32 (1)}) - 3 \frac{{(k^{8})}^{4}}{2^{4}}$

Step 16.7.4.2.2

Cancel the common factor.

$- 3 (\frac{1}{k^{8}} \cdot \frac{32 (2 k^{40})}{32 \cdot 1}) - 3 \frac{{(k^{8})}^{4}}{2^{4}}$

Step 16.7.4.2.3

Rewrite the expression.

$- 3 (\frac{1}{k^{8}} \cdot \frac{2 k^{40}}{1}) - 3 \frac{{(k^{8})}^{4}}{2^{4}}$

Step 16.7.4.2.4

Divide $2 k^{40}$ by $1$ .

$- 3 (\frac{1}{k^{8}} (2 k^{40})) - 3 \frac{{(k^{8})}^{4}}{2^{4}}$

Step 16.7.5

Combine $2$ and $\frac{1}{k^{8}}$ .

$- 3 (\frac{2}{k^{8}} k^{40}) - 3 \frac{{(k^{8})}^{4}}{2^{4}}$

Step 16.7.6

Combine $\frac{2}{k^{8}}$ and $k^{40}$ .

$- 3 \frac{2 k^{40}}{k^{8}} - 3 \frac{{(k^{8})}^{4}}{2^{4}}$

Step 16.7.7

Cancel the common factor of $k^{40}$ and $k^{8}$ .

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Step 16.7.7.1

Factor $k^{8}$ out of $2 k^{40}$ .

$- 3 \frac{k^{8} (2 k^{32})}{k^{8}} - 3 \frac{{(k^{8})}^{4}}{2^{4}}$

Step 16.7.7.2

Cancel the common factors.

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Step 16.7.7.2.1

Multiply by $1$ .

$- 3 \frac{k^{8} (2 k^{32})}{k^{8} \cdot 1} - 3 \frac{{(k^{8})}^{4}}{2^{4}}$

Step 16.7.7.2.2

Cancel the common factor.

$- 3 \frac{k^{8} (2 k^{32})}{k^{8} \cdot 1} - 3 \frac{{(k^{8})}^{4}}{2^{4}}$

Step 16.7.7.2.3

Rewrite the expression.

$- 3 \frac{2 k^{32}}{1} - 3 \frac{{(k^{8})}^{4}}{2^{4}}$

Step 16.7.7.2.4

Divide $2 k^{32}$ by $1$ .

$- 3 (2 k^{32}) - 3 \frac{{(k^{8})}^{4}}{2^{4}}$

Step 16.7.8

Multiply $2$ by $- 3$ .

$- 6 k^{32} - 3 \frac{{(k^{8})}^{4}}{2^{4}}$

Step 16.7.9

Multiply the exponents in ${(k^{8})}^{4}$ .

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Step 16.7.9.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- 6 k^{32} - 3 \frac{k^{8 \cdot 4}}{2^{4}}$

Step 16.7.9.2

Multiply $8$ by $4$ .

$- 6 k^{32} - 3 \frac{k^{32}}{2^{4}}$

Step 16.7.10

Raise $2$ to the power of $4$ .

$- 6 k^{32} - 3 \frac{k^{32}}{16}$

Step 16.7.11

Combine $- 3$ and $\frac{k^{32}}{16}$ .

$- 6 k^{32} + \frac{- 3 k^{32}}{16}$

Step 16.7.12

Move the negative in front of the fraction.

$- 6 k^{32} - \frac{3 k^{32}}{16}$

Step 16.7.13

To write $- 6 k^{32}$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$- 6 k^{32} \cdot \frac{16}{16} - \frac{3 k^{32}}{16}$

Step 16.7.14

Combine $- 6 k^{32}$ and $\frac{16}{16}$ .

$\frac{- 6 k^{32} \cdot 16}{16} - \frac{3 k^{32}}{16}$

Step 16.7.15

Combine the numerators over the common denominator.

$\frac{- 6 k^{32} \cdot 16 - 3 k^{32}}{16}$

Step 16.7.16

Multiply $16$ by $- 6$ .

$\frac{- 96 k^{32} - 3 k^{32}}{16}$

Step 16.7.17

Subtract $3 k^{32}$ from $- 96 k^{32}$ .

$\frac{- 99 k^{32}}{16}$

Step 16.7.18

Move the negative in front of the fraction.

$- \frac{99 k^{32}}{16}$