Calculus Examples

$p (t) = (\frac{1}{5255}) \cdot 2^{t} + 2 \cdot 10^{8}$

Step 1

Differentiate using the Sum Rule.

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Step 1.1

Raise $10$ to the power of $8$ .

$\frac{d}{d t} [(\frac{1}{5255}) \cdot 2^{t} + 2 \cdot 100000000]$

Step 1.2

By the Sum Rule, the derivative of $(\frac{1}{5255}) \cdot 2^{t} + 2 \cdot 100000000$ with respect to $t$ is $\frac{d}{d t} [(\frac{1}{5255}) \cdot 2^{t}] + \frac{d}{d t} [2 \cdot 100000000]$ .

$\frac{d}{d t} [(\frac{1}{5255}) \cdot 2^{t}] + \frac{d}{d t} [2 \cdot 100000000]$

Step 2

Evaluate $\frac{d}{d t} [(\frac{1}{5255}) \cdot 2^{t}]$ .

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Step 2.1

Combine $\frac{1}{5255}$ and $2^{t}$ .

$\frac{d}{d t} [\frac{2^{t}}{5255}] + \frac{d}{d t} [2 \cdot 100000000]$

Step 2.2

Since $\frac{1}{5255}$ is constant with respect to $t$ , the derivative of $\frac{2^{t}}{5255}$ with respect to $t$ is $\frac{1}{5255} \frac{d}{d t} [2^{t}]$ .

$\frac{1}{5255} \frac{d}{d t} [2^{t}] + \frac{d}{d t} [2 \cdot 100000000]$

Step 2.3

Differentiate using the Exponential Rule which states that $\frac{d}{d t} [a^{t}]$ is $a^{t} \ln (a)$ where $a$ = $2$ .

$\frac{1}{5255} (2^{t} \ln (2)) + \frac{d}{d t} [2 \cdot 100000000]$

Step 2.4

Combine $2^{t}$ and $\frac{1}{5255}$ .

$\frac{2^{t}}{5255} \ln (2) + \frac{d}{d t} [2 \cdot 100000000]$

Step 2.5

Combine $\frac{2^{t}}{5255}$ and $\ln (2)$ .

$\frac{2^{t} \ln (2)}{5255} + \frac{d}{d t} [2 \cdot 100000000]$

Step 3

Evaluate $\frac{d}{d t} [2 \cdot 100000000]$ .

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Step 3.1

Multiply $2$ by $100000000$ .

$\frac{2^{t} \ln (2)}{5255} + \frac{d}{d t} [200000000]$

Step 3.2

Since $200000000$ is constant with respect to $t$ , the derivative of $200000000$ with respect to $t$ is $0$ .

$\frac{2^{t} \ln (2)}{5255} + 0$

Step 4

Add $\frac{2^{t} \ln (2)}{5255}$ and $0$ .

$\frac{2^{t} \ln (2)}{5255}$