Calculus Examples

$f (x) = 200000 (1) {(0.9)}^{t} \ln (200000)$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Multiply $200000$ by $1$ .

$\frac{d}{d t} [200000 {(0.9)}^{t} \ln (200000)]$

Step 1.2

Since $200000 \ln (200000)$ is constant with respect to $t$ , the derivative of $200000 \cdot {0.9}^{t} \ln (200000)$ with respect to $t$ is $200000 \ln (200000) \frac{d}{d t} [{0.9}^{t}]$ .

$200000 \ln (200000) \frac{d}{d t} [{0.9}^{t}]$

Step 2

Differentiate using the Exponential Rule which states that $\frac{d}{d t} [a^{t}]$ is $a^{t} \ln (a)$ where $a$ = $0.9$ .

$200000 \ln (200000) ({0.9}^{t} \ln (0.9))$

Step 3

Reorder the factors of $200000 \ln (200000) \cdot {0.9}^{t} \ln (0.9)$ .

$200000 \cdot {0.9}^{t} \ln (0.9) \ln (200000)$