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

$0.000001$ .

$8.96548404$