Calculus Examples

$y = \ln (t^{10})$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d t} (y) = \frac{d}{d t} (\ln (t^{10}))$

Step 2

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

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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 $g (t) = t^{10}$ .

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Step 3.1.1

To apply the Chain Rule, set $u$ as $t^{10}$ .

$\frac{d}{d u} [\ln (u)] \frac{d}{d t} [t^{10}]$

Step 3.1.2

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

$\frac{1}{u} \frac{d}{d t} [t^{10}]$

Step 3.1.3

Replace all occurrences of $u$ with $t^{10}$ .

$\frac{1}{t^{10}} \frac{d}{d t} [t^{10}]$

Step 3.2

Differentiate using the Power Rule.

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Step 3.2.1

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 10$ .

$\frac{1}{t^{10}} (10 t^{9})$

Step 3.2.2

Simplify terms.

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Step 3.2.2.1

Combine $10$ and $\frac{1}{t^{10}}$ .

$\frac{10}{t^{10}} t^{9}$

Step 3.2.2.2

Combine $\frac{10}{t^{10}}$ and $t^{9}$ .

$\frac{10 t^{9}}{t^{10}}$

Step 3.2.2.3

Cancel the common factor of $t^{9}$ and $t^{10}$ .

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Step 3.2.2.3.1

Factor $t^{9}$ out of $10 t^{9}$ .

$\frac{t^{9} \cdot 10}{t^{10}}$

Step 3.2.2.3.2

Cancel the common factors.

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Step 3.2.2.3.2.1

Factor $t^{9}$ out of $t^{10}$ .

$\frac{t^{9} \cdot 10}{t^{9} t}$

Step 3.2.2.3.2.2

Cancel the common factor.

$\frac{t^{9} \cdot 10}{t^{9} t}$

Step 3.2.2.3.2.3

Rewrite the expression.

$\frac{10}{t}$

Step 4

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

$y' = \frac{10}{t}$

Step 5

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

$\frac{d y}{d t} = \frac{10}{t}$