Calculus Examples

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

Step 1

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 $f (t) = 1 + \ln (t)$ and $g (t) = 1 - \ln (t)$ .

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

Since $1$ is constant with respect to $t$ , the derivative of $1$ with respect to $t$ is $0$ .

$\frac{(1 - \ln (t)) (0 + \frac{d}{d t} [\ln (t)]) - (1 + \ln (t)) \frac{d}{d t} [1 - \ln (t)]}{{(1 - \ln (t))}^{2}}$

Step 2.3

Add $0$ and $\frac{d}{d t} [\ln (t)]$ .

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

Step 3

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

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

Step 4

Differentiate.

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Step 4.1

By the Sum Rule, the derivative of $1 - \ln (t)$ with respect to $t$ is $\frac{d}{d t} [1] + \frac{d}{d t} [- \ln (t)]$ .

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

Step 4.2

Since $1$ is constant with respect to $t$ , the derivative of $1$ with respect to $t$ is $0$ .

$\frac{(1 - \ln (t)) \frac{1}{t} - (1 + \ln (t)) (0 + \frac{d}{d t} [- \ln (t)])}{{(1 - \ln (t))}^{2}}$

Step 4.3

Add $0$ and $\frac{d}{d t} [- \ln (t)]$ .

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

Step 4.4

Since $- 1$ is constant with respect to $t$ , the derivative of $- \ln (t)$ with respect to $t$ is $- \frac{d}{d t} [\ln (t)]$ .

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

Step 4.5

Multiply.

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Step 4.5.1

Multiply $- 1$ by $- 1$ .

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

Step 4.5.2

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

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

Step 5

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

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

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Simplify the numerator.

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Step 6.3.1

Combine the opposite terms in $1 \frac{1}{t} - \ln (t) \frac{1}{t} + 1 \frac{1}{t} + \ln (t) \frac{1}{t}$ .

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Step 6.3.1.1

Add $- \ln (t) \frac{1}{t}$ and $\ln (t) \frac{1}{t}$ .

$\frac{1 \frac{1}{t} + 1 \frac{1}{t} + 0}{{(1 - \ln (t))}^{2}}$

Step 6.3.1.2

Add $1 \frac{1}{t} + 1 \frac{1}{t}$ and $0$ .

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

Step 6.3.2

Simplify each term.

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Step 6.3.2.1

Multiply $\frac{1}{t}$ by $1$ .

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

Step 6.3.2.2

Multiply $\frac{1}{t}$ by $1$ .

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

Step 6.3.3

Combine the numerators over the common denominator.

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

Step 6.3.4

Add $1$ and $1$ .

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

Step 6.4

Combine terms.

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Step 6.4.1

Rewrite $\frac{\frac{2}{t}}{{(1 - \ln (t))}^{2}}$ as a product.

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

Step 6.4.2

Multiply $\frac{2}{t}$ by $\frac{1}{{(1 - \ln (t))}^{2}}$ .

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