Calculus Examples

$\ln (t \cdot 5^{t})$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = \ln (t)$ and $g (t) = t \cdot 5^{t}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $t \cdot 5^{t}$ .

$\frac{d}{d u} [\ln (u)] \frac{d}{d t} [t \cdot 5^{t}]$

Step 1.2

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

$\frac{1}{u} \frac{d}{d t} [t \cdot 5^{t}]$

Step 1.3

Replace all occurrences of $u$ with $t \cdot 5^{t}$ .

$\frac{1}{t \cdot 5^{t}} \frac{d}{d t} [t \cdot 5^{t}]$

Step 2

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 $f (t) = t$ and $g (t) = 5^{t}$ .

$\frac{1}{t \cdot 5^{t}} (t \frac{d}{d t} [5^{t}] + 5^{t} \frac{d}{d t} [t])$

Step 3

Differentiate using the Exponential Rule which states that $\frac{d}{d t} [a^{t}]$ is $a^{t} \ln (a)$ where $a$ = $5$ .

$\frac{1}{t \cdot 5^{t}} (t (5^{t} \ln (5)) + 5^{t} \frac{d}{d t} [t])$

Step 4

Differentiate using the Power Rule.

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Step 4.1

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 1$ .

$\frac{1}{t \cdot 5^{t}} (t (5^{t} \ln (5)) + 5^{t} \cdot 1)$

Step 4.2

Multiply $5^{t}$ by $1$ .

$\frac{1}{t \cdot 5^{t}} (t (5^{t} \ln (5)) + 5^{t})$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$\frac{1}{t \cdot 5^{t}} (t (5^{t} \ln (5))) + \frac{1}{t \cdot 5^{t}} \cdot 5^{t}$

Step 5.2

Combine terms.

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Step 5.2.1

Combine $t$ and $\frac{1}{t \cdot 5^{t}}$ .

$\frac{t}{t \cdot 5^{t}} (5^{t} \ln (5)) + \frac{1}{t \cdot 5^{t}} \cdot 5^{t}$

Step 5.2.2

Combine $5^{t}$ and $\frac{t}{t \cdot 5^{t}}$ .

$\frac{5^{t} t}{t \cdot 5^{t}} \ln (5) + \frac{1}{t \cdot 5^{t}} \cdot 5^{t}$

Step 5.2.3

Combine $\frac{5^{t} t}{t \cdot 5^{t}}$ and $\ln (5)$ .

$\frac{5^{t} t \ln (5)}{t \cdot 5^{t}} + \frac{1}{t \cdot 5^{t}} \cdot 5^{t}$

Step 5.2.4

Cancel the common factor of $5^{t}$ .

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Step 5.2.4.1

Cancel the common factor.

$\frac{5^{t} t \ln (5)}{t \cdot 5^{t}} + \frac{1}{t \cdot 5^{t}} \cdot 5^{t}$

Step 5.2.4.2

Rewrite the expression.

$\frac{t \ln (5)}{t} + \frac{1}{t \cdot 5^{t}} \cdot 5^{t}$

Step 5.2.5

Cancel the common factor of $t$ .

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Step 5.2.5.1

Cancel the common factor.

$\frac{t \ln (5)}{t} + \frac{1}{t \cdot 5^{t}} \cdot 5^{t}$

Step 5.2.5.2

Divide $\ln (5)$ by $1$ .

$\ln (5) + \frac{1}{t \cdot 5^{t}} \cdot 5^{t}$

Step 5.2.6

Combine $\frac{1}{t \cdot 5^{t}}$ and $5^{t}$ .

$\ln (5) + \frac{5^{t}}{t \cdot 5^{t}}$

Step 5.2.7

Cancel the common factor of $5^{t}$ .

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Step 5.2.7.1

Cancel the common factor.

$\ln (5) + \frac{5^{t}}{t \cdot 5^{t}}$

Step 5.2.7.2

Rewrite the expression.

$\ln (5) + \frac{1}{t}$

Step 5.3

Reorder terms.

$\frac{1}{t} + \ln (5)$