Calculus Examples

$y = t^{2} \ln^{4} (t)$

Step 1

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 $f (t) = t^{2}$ and $g (t) = \ln^{4} (t)$ .

$t^{2} \frac{d}{d t} [\ln^{4} (t)] + \ln^{4} (t) \frac{d}{d t} [t^{2}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{4}$ and $g (t) = \ln (t)$ .

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Step 2.1

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

$t^{2} (\frac{d}{d u} [u^{4}] \frac{d}{d t} [\ln (t)]) + \ln^{4} (t) \frac{d}{d t} [t^{2}]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 4$ .

$t^{2} (4 u^{3} \frac{d}{d t} [\ln (t)]) + \ln^{4} (t) \frac{d}{d t} [t^{2}]$

Step 2.3

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

$t^{2} (4 \ln^{3} (t) \frac{d}{d t} [\ln (t)]) + \ln^{4} (t) \frac{d}{d t} [t^{2}]$

Step 3

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

$t^{2} (4 \ln^{3} (t) \frac{1}{t}) + \ln^{4} (t) \frac{d}{d t} [t^{2}]$

Step 4

Differentiate using the Power Rule.

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Step 4.1

Combine $\frac{1}{t}$ and $4$ .

$t^{2} (\frac{4}{t} \ln^{3} (t)) + \ln^{4} (t) \frac{d}{d t} [t^{2}]$

Step 4.2

Simplify terms.

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Step 4.2.1

Combine $\frac{4}{t}$ and $\ln^{3} (t)$ .

$t^{2} \frac{4 \ln^{3} (t)}{t} + \ln^{4} (t) \frac{d}{d t} [t^{2}]$

Step 4.2.2

Combine $t^{2}$ and $\frac{4 \ln^{3} (t)}{t}$ .

$\frac{t^{2} (4 \ln^{3} (t))}{t} + \ln^{4} (t) \frac{d}{d t} [t^{2}]$

Step 4.2.3

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

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Step 4.2.3.1

Factor $t$ out of $t^{2} (4 \ln^{3} (t))$ .

$\frac{t (t (4 \ln^{3} (t)))}{t} + \ln^{4} (t) \frac{d}{d t} [t^{2}]$

Step 4.2.3.2

Cancel the common factors.

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Step 4.2.3.2.1

Raise $t$ to the power of $1$ .

$\frac{t (t (4 \ln^{3} (t)))}{t^{1}} + \ln^{4} (t) \frac{d}{d t} [t^{2}]$

Step 4.2.3.2.2

Factor $t$ out of $t^{1}$ .

$\frac{t (t (4 \ln^{3} (t)))}{t \cdot 1} + \ln^{4} (t) \frac{d}{d t} [t^{2}]$

Step 4.2.3.2.3

Cancel the common factor.

$\frac{t (t (4 \ln^{3} (t)))}{t \cdot 1} + \ln^{4} (t) \frac{d}{d t} [t^{2}]$

Step 4.2.3.2.4

Rewrite the expression.

$\frac{t (4 \ln^{3} (t))}{1} + \ln^{4} (t) \frac{d}{d t} [t^{2}]$

Step 4.2.3.2.5

Divide $t (4 \ln^{3} (t))$ by $1$ .

$t (4 \ln^{3} (t)) + \ln^{4} (t) \frac{d}{d t} [t^{2}]$

Step 4.3

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

$t (4 \ln^{3} (t)) + \ln^{4} (t) (2 t)$

Step 4.4

Reorder terms.

$4 t \ln^{3} (t) + 2 t \ln^{4} (t)$