Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Differentiate both sides of the equation.

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

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

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

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

Step 4.2

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{2}$ and $g (t) = \ln (4 t)$ .

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Step 4.2.1

To apply the Chain Rule, set $u_{1}$ as $\ln (4 t)$ .

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

Step 4.2.2

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

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

Step 4.2.3

Replace all occurrences of $u_{1}$ with $\ln (4 t)$ .

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

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 $f (t) = \ln (t)$ and $g (t) = 4 t$ .

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Step 4.3.1

To apply the Chain Rule, set $u_{2}$ as $4 t$ .

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

Step 4.3.2

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

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

Step 4.3.3

Replace all occurrences of $u_{2}$ with $4 t$ .

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

Step 4.4

Differentiate.

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Step 4.4.1

Combine $\frac{1}{4 t}$ and $2$ .

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

Step 4.4.2

Simplify terms.

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Step 4.4.2.1

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

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

Step 4.4.2.2

Cancel the common factor of $2$ and $4$ .

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Step 4.4.2.2.1

Factor $2$ out of $2 \ln (4 t)$ .

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

Step 4.4.2.2.2

Cancel the common factors.

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Step 4.4.2.2.2.1

Factor $2$ out of $4 t$ .

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

Step 4.4.2.2.2.2

Cancel the common factor.

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

Step 4.4.2.2.2.3

Rewrite the expression.

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

Step 4.4.2.3

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

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

Step 4.4.2.4

Cancel the common factor of $t$ .

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Step 4.4.2.4.1

Cancel the common factor.

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

Step 4.4.2.4.2

Rewrite the expression.

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

Step 4.4.3

Since $4$ is constant with respect to $t$ , the derivative of $4 t$ with respect to $t$ is $4 \frac{d}{d t} [t]$ .

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

Step 4.4.4

Simplify terms.

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Step 4.4.4.1

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

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

Step 4.4.4.2

Cancel the common factor of $4$ and $2$ .

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Step 4.4.4.2.1

Factor $2$ out of $4 \ln (4 t)$ .

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

Step 4.4.4.2.2

Cancel the common factors.

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Step 4.4.4.2.2.1

Factor $2$ out of $2$ .

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

Step 4.4.4.2.2.2

Cancel the common factor.

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

Step 4.4.4.2.2.3

Rewrite the expression.

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

Step 4.4.4.2.2.4

Divide $2 \ln (4 t)$ by $1$ .

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

Step 4.4.5

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

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

Step 4.4.6

Multiply $2$ by $1$ .

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

Step 4.4.7

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

$2 \ln (4 t) + \ln^{2} (4 t) \cdot 1$

Step 4.4.8

Simplify the expression.

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Step 4.4.8.1

Multiply $\ln^{2} (4 t)$ by $1$ .

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

Step 4.4.8.2

Reorder terms.

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

Step 5

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

$y' = \ln^{2} (4 t) + 2 \ln (4 t)$

Step 6

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

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