Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

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

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Step 3.1.1

To apply the Chain Rule, set $u_{1}$ as $\ln (2 t^{3})$ .

$\frac{d}{d u_{1}} [\ln (u_{1})] \frac{d}{d t} [\ln (2 t^{3})]$

Step 3.1.2

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

$\frac{1}{u_{1}} \frac{d}{d t} [\ln (2 t^{3})]$

Step 3.1.3

Replace all occurrences of $u_{1}$ with $\ln (2 t^{3})$ .

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

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

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Step 3.2.1

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

$\frac{1}{\ln (2 t^{3})} (\frac{d}{d u_{2}} [\ln (u_{2})] \frac{d}{d t} [2 t^{3}])$

Step 3.2.2

The derivative of $\ln (u_{2})$ with respect to $u_{2}$ is $\frac{1}{u_{2}}$ .

$\frac{1}{\ln (2 t^{3})} (\frac{1}{u_{2}} \frac{d}{d t} [2 t^{3}])$

Step 3.2.3

Replace all occurrences of $u_{2}$ with $2 t^{3}$ .

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

Step 3.3

Differentiate.

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Step 3.3.1

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

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

Step 3.3.2

Since $2$ is constant with respect to $t$ , the derivative of $2 t^{3}$ with respect to $t$ is $2 \frac{d}{d t} [t^{3}]$ .

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

Step 3.3.3

Simplify terms.

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Step 3.3.3.1

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

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

Step 3.3.3.2

Cancel the common factor of $2$ .

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Step 3.3.3.2.1

Cancel the common factor.

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

Step 3.3.3.2.2

Rewrite the expression.

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

Step 3.3.4

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

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

Step 3.3.5

Simplify terms.

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Step 3.3.5.1

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

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

Step 3.3.5.2

Combine $\frac{3}{t^{3} \ln (2 t^{3})}$ and $t^{2}$ .

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

Step 3.3.5.3

Cancel the common factor of $t^{2}$ and $t^{3}$ .

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Step 3.3.5.3.1

Factor $t^{2}$ out of $3 t^{2}$ .

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

Step 3.3.5.3.2

Cancel the common factors.

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Step 3.3.5.3.2.1

Factor $t^{2}$ out of $t^{3} \ln (2 t^{3})$ .

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

Step 3.3.5.3.2.2

Cancel the common factor.

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

Step 3.3.5.3.2.3

Rewrite the expression.

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

Step 4

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

$y' = \frac{3}{t \ln (2 t^{3})}$

Step 5

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

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