Calculus Examples

$y = \ln (t^{\frac{3}{2}})$

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 $g (t) = t^{\frac{3}{2}}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $t^{\frac{3}{2}}$ .

$\frac{d}{d u} [\ln (u)] \frac{d}{d t} [t^{\frac{3}{2}}]$

Step 1.2

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

$\frac{1}{u} \frac{d}{d t} [t^{\frac{3}{2}}]$

Step 1.3

Replace all occurrences of $u$ with $t^{\frac{3}{2}}$ .

$\frac{1}{t^{\frac{3}{2}}} \frac{d}{d t} [t^{\frac{3}{2}}]$

Step 2

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = \frac{3}{2}$ .

$\frac{1}{t^{\frac{3}{2}}} (\frac{3}{2} t^{\frac{3}{2} - 1})$

Step 3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{t^{\frac{3}{2}}} (\frac{3}{2} t^{\frac{3}{2} - 1 \cdot \frac{2}{2}})$

Step 4

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

$\frac{1}{t^{\frac{3}{2}}} (\frac{3}{2} t^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}})$

Step 5

Combine the numerators over the common denominator.

$\frac{1}{t^{\frac{3}{2}}} (\frac{3}{2} t^{\frac{3 - 1 \cdot 2}{2}})$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

$\frac{1}{t^{\frac{3}{2}}} (\frac{3}{2} t^{\frac{3 - 2}{2}})$

Step 6.2

Subtract $2$ from $3$ .

$\frac{1}{t^{\frac{3}{2}}} (\frac{3}{2} t^{\frac{1}{2}})$

Step 7

Combine $\frac{3}{2}$ and $t^{\frac{1}{2}}$ .

$\frac{1}{t^{\frac{3}{2}}} \cdot \frac{3 t^{\frac{1}{2}}}{2}$

Step 8

Multiply $\frac{1}{t^{\frac{3}{2}}}$ by $\frac{3 t^{\frac{1}{2}}}{2}$ .

$\frac{3 t^{\frac{1}{2}}}{t^{\frac{3}{2}} \cdot 2}$

Step 9

Simplify the expression.

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Step 9.1

Move $2$ to the left of $t^{\frac{3}{2}}$ .

$\frac{3 t^{\frac{1}{2}}}{2 \cdot t^{\frac{3}{2}}}$

Step 9.2

Move $t^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{3}{2 t^{\frac{3}{2}} t^{- \frac{1}{2}}}$

Step 10

Simplify the denominator.

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Step 10.1

Multiply $t^{\frac{3}{2}}$ by $t^{- \frac{1}{2}}$ by adding the exponents.

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Step 10.1.1

Move $t^{- \frac{1}{2}}$ .

$\frac{3}{2 (t^{- \frac{1}{2}} t^{\frac{3}{2}})}$

Step 10.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{3}{2 t^{- \frac{1}{2} + \frac{3}{2}}}$

Step 10.1.3

Combine the numerators over the common denominator.

$\frac{3}{2 t^{\frac{- 1 + 3}{2}}}$

Step 10.1.4

Add $- 1$ and $3$ .

$\frac{3}{2 t^{\frac{2}{2}}}$

Step 10.1.5

Divide $2$ by $2$ .

$\frac{3}{2 t^{1}}$

Step 10.2

Simplify $2 t^{1}$ .

$\frac{3}{2 t}$