Calculus Examples

$(45 + e^{t}) \ln (t)$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d t} [f (t) g (t)]$ is $f (t) \frac{d}{d t} [g (t)] + g (t) \frac{d}{d t} [f (t)]$ where $f (t) = 45 + e^{t}$ and $g (t) = \ln (t)$ .

$(45 + e^{t}) \frac{d}{d t} [\ln (t)] + \ln (t) \frac{d}{d t} [45 + e^{t}]$

Step 2

The derivative of $\ln (t)$ with respect to $t$ is $\frac{1}{t}$ .

$(45 + e^{t}) \frac{1}{t} + \ln (t) \frac{d}{d t} [45 + e^{t}]$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $45 + e^{t}$ with respect to $t$ is $\frac{d}{d t} [45] + \frac{d}{d t} [e^{t}]$ .

$(45 + e^{t}) \frac{1}{t} + \ln (t) (\frac{d}{d t} [45] + \frac{d}{d t} [e^{t}])$

Step 3.2

Since $45$ is constant with respect to $t$ , the derivative of $45$ with respect to $t$ is $0$ .

$(45 + e^{t}) \frac{1}{t} + \ln (t) (0 + \frac{d}{d t} [e^{t}])$

Step 3.3

Add $0$ and $\frac{d}{d t} [e^{t}]$ .

$(45 + e^{t}) \frac{1}{t} + \ln (t) \frac{d}{d t} [e^{t}]$

Step 4

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

$(45 + e^{t}) \frac{1}{t} + \ln (t) e^{t}$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$45 \frac{1}{t} + e^{t} \frac{1}{t} + \ln (t) e^{t}$

Step 5.2

Combine terms.

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Step 5.2.1

Combine $45$ and $\frac{1}{t}$ .

$\frac{45}{t} + e^{t} \frac{1}{t} + \ln (t) e^{t}$

Step 5.2.2

Combine $e^{t}$ and $\frac{1}{t}$ .

$\frac{45}{t} + \frac{e^{t}}{t} + \ln (t) e^{t}$

Step 5.3

Reorder terms.

$e^{t} \ln (t) + \frac{45}{t} + \frac{e^{t}}{t}$