Calculus Examples

$f (x) = \frac{2 \ln (5) \cdot 5^{- 3}}{7} - \frac{3 {(5)}^{- 3}}{7}$

Step 1

Move $5^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{d}{d x} [\frac{2 \ln (5)}{7 \cdot 5^{3}} - \frac{3 {(5)}^{- 3}}{7}]$

Step 2

Simplify $2 \ln (5)$ by moving $2$ inside the logarithm.

$\frac{d}{d x} [\frac{\ln (5^{2})}{7 \cdot 5^{3}} - \frac{3 {(5)}^{- 3}}{7}]$

Step 3

Raise $5$ to the power of $3$ .

$\frac{d}{d x} [\frac{\ln (5^{2})}{7 \cdot 125} - \frac{3 {(5)}^{- 3}}{7}]$

Step 4

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

$\frac{d}{d x} [\frac{\ln (25)}{7 \cdot 125} - \frac{3 {(5)}^{- 3}}{7}]$

Step 5

Multiply $7$ by $125$ .

$\frac{d}{d x} [\frac{\ln (25)}{875} - \frac{3 {(5)}^{- 3}}{7}]$

Step 6

Rewrite $\frac{\ln (25)}{875}$ as $\frac{1}{875} \ln (25)$ .

$\frac{d}{d x} [\frac{1}{875} \ln (25) - \frac{3 {(5)}^{- 3}}{7}]$

Step 7

Simplify $\frac{1}{875} \ln (25)$ by moving $\frac{1}{875}$ inside the logarithm.

$\frac{d}{d x} [\ln (25^{\frac{1}{875}}) - \frac{3 {(5)}^{- 3}}{7}]$

Step 8

Simplify the expression.

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Step 8.1

Move ${(5)}^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{d}{d x} [\ln (25^{\frac{1}{875}}) - \frac{3}{7 \cdot 5^{3}}]$

Step 8.2

Raise $5$ to the power of $3$ .

$\frac{d}{d x} [\ln (25^{\frac{1}{875}}) - \frac{3}{7 \cdot 125}]$

Step 8.3

Multiply $7$ by $125$ .

$\frac{d}{d x} [\ln (25^{\frac{1}{875}}) - \frac{3}{875}]$

Step 9

By the Sum Rule, the derivative of $\ln (25^{\frac{1}{875}}) - \frac{3}{875}$ with respect to $x$ is $\frac{d}{d x} [\ln (25^{\frac{1}{875}})] + \frac{d}{d x} [- \frac{3}{875}]$ .

$\frac{d}{d x} [\ln (25^{\frac{1}{875}})] + \frac{d}{d x} [- \frac{3}{875}]$

Step 10

Since $\ln (25^{\frac{1}{875}})$ is constant with respect to $x$ , the derivative of $\ln (25^{\frac{1}{875}})$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- \frac{3}{875}]$

Step 11

Since $- \frac{3}{875}$ is constant with respect to $x$ , the derivative of $- \frac{3}{875}$ with respect to $x$ is $0$ .

$0 + 0$

Step 12

Add $0$ and $0$ .

$0$