Calculus Examples

$B = P {(1 + \frac{r}{100})}^{t}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d t} (B) = \frac{d}{d t} (P {(1 + \frac{r}{100})}^{t})$

Step 2

The derivative of $B$ with respect to $t$ is $B'$ .

$B'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Since $P$ is constant with respect to $t$ , the derivative of $P {(1 + \frac{r}{100})}^{t}$ with respect to $t$ is $P \frac{d}{d t} [{(1 + \frac{r}{100})}^{t}]$ .

$P \frac{d}{d t} [{(1 + \frac{r}{100})}^{t}]$

Step 3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d t} [a^{t}]$ is $a^{t} \ln (a)$ where $a$ = $1 + \frac{r}{100}$ .

$P {(1 + \frac{r}{100})}^{t} \ln (1 + \frac{r}{100})$

Step 4

Reform the equation by setting the left side equal to the right side.

$B' = P {(1 + \frac{r}{100})}^{t} \ln (1 + \frac{r}{100})$

Step 5

Replace $B'$ with $\frac{d B}{d t}$ .

$\frac{d B}{d t} = P {(1 + \frac{r}{100})}^{t} \ln (1 + \frac{r}{100})$