Calculus Examples

$\frac{7 t}{\ln (t)}$

Step 1

Since $7$ is constant with respect to $t$ , the derivative of $\frac{7 t}{\ln (t)}$ with respect to $t$ is $7 \frac{d}{d t} [\frac{t}{\ln (t)}]$ .

$7 \frac{d}{d t} [\frac{t}{\ln (t)}]$

Step 2

Differentiate using the Quotient Rule which states that $\frac{d}{d t} [\frac{f (t)}{g (t)}]$ is $\frac{g (t) \frac{d}{d t} [f (t)] - f (t) \frac{d}{d t} [g (t)]}{{g (t)}^{2}}$ where $f (t) = t$ and $g (t) = \ln (t)$ .

$7 \frac{\ln (t) \frac{d}{d t} [t] - t \frac{d}{d t} [\ln (t)]}{\ln^{2} (t)}$

Step 3

Differentiate using the Power Rule.

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Step 3.1

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

$7 \frac{\ln (t) \cdot 1 - t \frac{d}{d t} [\ln (t)]}{\ln^{2} (t)}$

Step 3.2

Multiply $\ln (t)$ by $1$ .

$7 \frac{\ln (t) - t \frac{d}{d t} [\ln (t)]}{\ln^{2} (t)}$

Step 4

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

$7 \frac{\ln (t) - t \frac{1}{t}}{\ln^{2} (t)}$

Step 5

Simplify terms.

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Step 5.1

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

$7 \frac{\ln (t) - \frac{t}{t}}{\ln^{2} (t)}$

Step 5.2

Cancel the common factor of $t$ .

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Step 5.2.1

Cancel the common factor.

$7 \frac{\ln (t) - \frac{t}{t}}{\ln^{2} (t)}$

Step 5.2.2

Rewrite the expression.

$7 \frac{\ln (t) - 1 \cdot 1}{\ln^{2} (t)}$

Step 5.3

Multiply $- 1$ by $1$ .

$7 \frac{\ln (t) - 1}{\ln^{2} (t)}$

Step 5.4

Combine $7$ and $\frac{\ln (t) - 1}{\ln^{2} (t)}$ .

$\frac{7 (\ln (t) - 1)}{\ln^{2} (t)}$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$\frac{7 \ln (t) + 7 \cdot - 1}{\ln^{2} (t)}$

Step 6.2

Simplify each term.

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Step 6.2.1

Simplify $7 \ln (t)$ by moving $7$ inside the logarithm.

$\frac{\ln (t^{7}) + 7 \cdot - 1}{\ln^{2} (t)}$

Step 6.2.2

Multiply $7$ by $- 1$ .

$\frac{\ln (t^{7}) - 7}{\ln^{2} (t)}$