Calculus Examples

$z = \ln (3 t - t)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d t} (z) = \frac{d}{d t} (\ln (3 t - t))$

Step 2

The derivative of $z$ with respect to $t$ is $z'$ .

$z'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Subtract $t$ from $3 t$ .

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

Step 3.2

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 $g (t) = 2 t$ .

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Step 3.2.1

To apply the Chain Rule, set $u$ as $2 t$ .

$\frac{d}{d u} [\ln (u)] \frac{d}{d t} [2 t]$

Step 3.2.2

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

$\frac{1}{u} \frac{d}{d t} [2 t]$

Step 3.2.3

Replace all occurrences of $u$ with $2 t$ .

$\frac{1}{2 t} \frac{d}{d t} [2 t]$

Step 3.3

Differentiate.

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Step 3.3.1

Since $2$ is constant with respect to $t$ , the derivative of $2 t$ with respect to $t$ is $2 \frac{d}{d t} [t]$ .

$\frac{1}{2 t} (2 \frac{d}{d t} [t])$

Step 3.3.2

Simplify terms.

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Step 3.3.2.1

Combine $2$ and $\frac{1}{2 t}$ .

$\frac{2}{2 t} \frac{d}{d t} [t]$

Step 3.3.2.2

Cancel the common factor of $2$ .

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Step 3.3.2.2.1

Cancel the common factor.

$\frac{2}{2 t} \frac{d}{d t} [t]$

Step 3.3.2.2.2

Rewrite the expression.

$\frac{1}{t} \frac{d}{d t} [t]$

Step 3.3.3

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

$\frac{1}{t} \cdot 1$

Step 3.3.4

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

$\frac{1}{t}$

Step 4

Reform the equation by setting the left side equal to the right side.

$z' = \frac{1}{t}$

Step 5

Replace $z'$ with $\frac{d z}{d t}$ .

$\frac{d z}{d t} = \frac{1}{t}$