Calculus Examples

$\frac{8 {(2.000001)}^{\ln (2.000001)} - 8 {(2)}^{\ln (2)}}{0.000001}$

Step 1

Multiply $- 8 \cdot 2^{\ln (2)}$ .

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Step 1.1

Factor out negative.

$\frac{8 \cdot {2.000001}^{\ln (2.000001)} - (8 \cdot 2^{\ln (2)})}{0.000001}$

Step 1.2

Rewrite $8$ as $2^{3}$ .

$\frac{8 \cdot {2.000001}^{\ln (2.000001)} - (2^{3} \cdot 2^{\ln (2)})}{0.000001}$

Step 1.3

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

$\frac{8 \cdot {2.000001}^{\ln (2.000001)} - 2^{3 + \ln (2)}}{0.000001}$

Step 2

Replace $e$ with an approximation.

$\frac{8 \cdot {2.000001}^{\log_{2.71828182} (2.000001)} - 2^{3 + \ln (2)}}{0.000001}$

Step 3

Log base $2.71828182$ of $2.000001$ is approximately $0.69314768$ .

$\frac{8 \cdot {2.000001}^{0.69314768} - 2^{3 + \ln (2)}}{0.000001}$

Step 4

Raise $2.000001$ to the power of $0.69314768$ .

$\frac{8 \cdot 1.61680779 - 2^{3 + \ln (2)}}{0.000001}$

Step 5

Multiply $8$ by $1.61680779$ .

$\frac{12.93446234 - 2^{3 + \ln (2)}}{0.000001}$

Step 6

Replace $e$ with an approximation.

$\frac{12.93446234 - 2^{3 + \log_{2.71828182} (2)}}{0.000001}$

Step 7

Log base $2.71828182$ of $2$ is approximately $0.69314718$ .

$\frac{12.93446234 - 2^{3 + 0.69314718}}{0.000001}$

Step 8

Add $3$ and $0.69314718$ .

$\frac{12.93446234 - 2^{3.69314718}}{0.000001}$

Step 9

Raise $2$ to the power of $3.69314718$ .

$\frac{12.93446234 - 1 \cdot 12.93445337}{0.000001}$

Step 10

Multiply $- 1$ by $12.93445337$ .

$\frac{12.93446234 - 12.93445337}{0.000001}$

Step 11

Subtract $12.93445337$ from $12.93446234$ .

$\frac{0.00000896}{0.000001}$

Step 12

Divide $0.00000896$ by $0.000001$ .

$8.96548404$