Calculus Examples

$y = t \ln^{2} (1 t)$

Step 1

Remove parentheses.

$y = t \ln^{2} (1 t)$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d t} (y) = \frac{d}{d t} (t \ln^{2} (1 t))$

Step 3

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

$y'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

Multiply $t$ by $1$ .

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

Step 4.2

Differentiate using the Product Rule which states that $\frac{d}{d t} [f (t) g (t)]$ is $f (t) \frac{d}{d t} [g (t)] + g (t) \frac{d}{d t} [f (t)]$ where $f (t) = t$ and $g (t) = \ln^{2} (t)$ .

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

Step 4.3

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{2}$ and $g (t) = \ln (t)$ .

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Step 4.3.1

To apply the Chain Rule, set $u$ as $\ln (t)$ .

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

Step 4.3.2

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

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

Step 4.3.3

Replace all occurrences of $u$ with $\ln (t)$ .

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

Step 4.4

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

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

Step 4.5

Differentiate using the Power Rule.

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Step 4.5.1

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

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

Step 4.5.2

Simplify terms.

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Step 4.5.2.1

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

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

Step 4.5.2.2

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

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

Step 4.5.2.3

Cancel the common factor of $t$ .

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Step 4.5.2.3.1

Cancel the common factor.

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

Step 4.5.2.3.2

Divide $2 \ln (t)$ by $1$ .

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

Step 4.5.3

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

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

Step 4.5.4

Simplify the expression.

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Step 4.5.4.1

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

$2 \ln (t) + \ln^{2} (t)$

Step 4.5.4.2

Reorder terms.

$\ln^{2} (t) + 2 \ln (t)$

Step 5

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

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

Step 6

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

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