Calculus Examples

$g (t) = t^{2} \ln (e^{2 t} + 8)$

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 $g (t) = \ln (e^{2 t} + 8)$ .

$t^{2} \frac{d}{d t} [\ln (e^{2 t} + 8)] + \ln (e^{2 t} + 8) \frac{d}{d t} [t^{2}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = \ln (t)$ and $g (t) = e^{2 t} + 8$ .

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Step 2.1

To apply the Chain Rule, set $u_{1}$ as $e^{2 t} + 8$ .

$t^{2} (\frac{d}{d u_{1}} [\ln (u_{1})] \frac{d}{d t} [e^{2 t} + 8]) + \ln (e^{2 t} + 8) \frac{d}{d t} [t^{2}]$

Step 2.2

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

$t^{2} (\frac{1}{u_{1}} \frac{d}{d t} [e^{2 t} + 8]) + \ln (e^{2 t} + 8) \frac{d}{d t} [t^{2}]$

Step 2.3

Replace all occurrences of $u_{1}$ with $e^{2 t} + 8$ .

$t^{2} (\frac{1}{e^{2 t} + 8} \frac{d}{d t} [e^{2 t} + 8]) + \ln (e^{2 t} + 8) \frac{d}{d t} [t^{2}]$

Step 3

Differentiate using the Sum Rule.

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Step 3.1

Combine $\frac{1}{e^{2 t} + 8}$ and $t^{2}$ .

$\frac{t^{2}}{e^{2 t} + 8} \frac{d}{d t} [e^{2 t} + 8] + \ln (e^{2 t} + 8) \frac{d}{d t} [t^{2}]$

Step 3.2

By the Sum Rule, the derivative of $e^{2 t} + 8$ with respect to $t$ is $\frac{d}{d t} [e^{2 t}] + \frac{d}{d t} [8]$ .

$\frac{t^{2}}{e^{2 t} + 8} (\frac{d}{d t} [e^{2 t}] + \frac{d}{d t} [8]) + \ln (e^{2 t} + 8) \frac{d}{d t} [t^{2}]$

Step 4

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

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Step 4.1

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

$\frac{t^{2}}{e^{2 t} + 8} (\frac{d}{d u_{2}} [e^{u_{2}}] \frac{d}{d t} [2 t] + \frac{d}{d t} [8]) + \ln (e^{2 t} + 8) \frac{d}{d t} [t^{2}]$

Step 4.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{2}} [a^{u_{2}}]$ is $a^{u_{2}} \ln (a)$ where $a$ = $e$ .

$\frac{t^{2}}{e^{2 t} + 8} (e^{u_{2}} \frac{d}{d t} [2 t] + \frac{d}{d t} [8]) + \ln (e^{2 t} + 8) \frac{d}{d t} [t^{2}]$

Step 4.3

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

$\frac{t^{2}}{e^{2 t} + 8} (e^{2 t} \frac{d}{d t} [2 t] + \frac{d}{d t} [8]) + \ln (e^{2 t} + 8) \frac{d}{d t} [t^{2}]$

Step 5

Differentiate.

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Step 5.1

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

$\frac{t^{2}}{e^{2 t} + 8} (e^{2 t} (2 \frac{d}{d t} [t]) + \frac{d}{d t} [8]) + \ln (e^{2 t} + 8) \frac{d}{d t} [t^{2}]$

Step 5.2

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

$\frac{t^{2}}{e^{2 t} + 8} (e^{2 t} (2 \cdot 1) + \frac{d}{d t} [8]) + \ln (e^{2 t} + 8) \frac{d}{d t} [t^{2}]$

Step 5.3

Simplify the expression.

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Step 5.3.1

Multiply $2$ by $1$ .

$\frac{t^{2}}{e^{2 t} + 8} (e^{2 t} \cdot 2 + \frac{d}{d t} [8]) + \ln (e^{2 t} + 8) \frac{d}{d t} [t^{2}]$

Step 5.3.2

Move $2$ to the left of $e^{2 t}$ .

$\frac{t^{2}}{e^{2 t} + 8} (2 \cdot e^{2 t} + \frac{d}{d t} [8]) + \ln (e^{2 t} + 8) \frac{d}{d t} [t^{2}]$

Step 5.4

Since $8$ is constant with respect to $t$ , the derivative of $8$ with respect to $t$ is $0$ .

$\frac{t^{2}}{e^{2 t} + 8} (2 e^{2 t} + 0) + \ln (e^{2 t} + 8) \frac{d}{d t} [t^{2}]$

Step 5.5

Combine fractions.

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Step 5.5.1

Add $2 e^{2 t}$ and $0$ .

$\frac{t^{2}}{e^{2 t} + 8} (2 e^{2 t}) + \ln (e^{2 t} + 8) \frac{d}{d t} [t^{2}]$

Step 5.5.2

Combine $2$ and $\frac{t^{2}}{e^{2 t} + 8}$ .

$\frac{2 t^{2}}{e^{2 t} + 8} e^{2 t} + \ln (e^{2 t} + 8) \frac{d}{d t} [t^{2}]$

Step 5.5.3

Combine $\frac{2 t^{2}}{e^{2 t} + 8}$ and $e^{2 t}$ .

$\frac{2 t^{2} e^{2 t}}{e^{2 t} + 8} + \ln (e^{2 t} + 8) \frac{d}{d t} [t^{2}]$

Step 5.6

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

$\frac{2 t^{2} e^{2 t}}{e^{2 t} + 8} + \ln (e^{2 t} + 8) (2 t)$

Step 6

To write $\ln (e^{2 t} + 8) (2 t)$ as a fraction with a common denominator, multiply by $\frac{e^{2 t} + 8}{e^{2 t} + 8}$ .

$\frac{2 t^{2} e^{2 t}}{e^{2 t} + 8} + \frac{\ln (e^{2 t} + 8) (2 t) (e^{2 t} + 8)}{e^{2 t} + 8}$

Step 7

Combine the numerators over the common denominator.

$\frac{2 t^{2} e^{2 t} + \ln (e^{2 t} + 8) (2 t) (e^{2 t} + 8)}{e^{2 t} + 8}$

Step 8

Simplify.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Simplify each term.

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Step 8.1.1.1

Rewrite using the commutative property of multiplication.

$\frac{2 t^{2} e^{2 t} + 2 \ln (e^{2 t} + 8) t (e^{2 t} + 8)}{e^{2 t} + 8}$

Step 8.1.1.2

Simplify $2 \ln (e^{2 t} + 8)$ by moving $2$ inside the logarithm.

$\frac{2 t^{2} e^{2 t} + \ln ({(e^{2 t} + 8)}^{2}) t (e^{2 t} + 8)}{e^{2 t} + 8}$

Step 8.1.1.3

Apply the distributive property.

$\frac{2 t^{2} e^{2 t} + \ln ({(e^{2 t} + 8)}^{2}) t e^{2 t} + \ln ({(e^{2 t} + 8)}^{2}) t \cdot 8}{e^{2 t} + 8}$

Step 8.1.1.4

Multiply $\ln ({(e^{2 t} + 8)}^{2}) t \cdot 8$ .

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Step 8.1.1.4.1

Reorder $\ln ({(e^{2 t} + 8)}^{2})$ and $8$ .

$\frac{2 t^{2} e^{2 t} + \ln ({(e^{2 t} + 8)}^{2}) t e^{2 t} + t \cdot (8 \cdot \ln ({(e^{2 t} + 8)}^{2}))}{e^{2 t} + 8}$

Step 8.1.1.4.2

Simplify $8 \cdot \ln ({(e^{2 t} + 8)}^{2})$ by moving $8$ inside the logarithm.

$\frac{2 t^{2} e^{2 t} + \ln ({(e^{2 t} + 8)}^{2}) t e^{2 t} + t \cdot \ln ({({(e^{2 t} + 8)}^{2})}^{8})}{e^{2 t} + 8}$

Step 8.1.1.5

Multiply the exponents in ${({(e^{2 t} + 8)}^{2})}^{8}$ .

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Step 8.1.1.5.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{2 t^{2} e^{2 t} + \ln ({(e^{2 t} + 8)}^{2}) t e^{2 t} + t \cdot \ln ({(e^{2 t} + 8)}^{2 \cdot 8})}{e^{2 t} + 8}$

Step 8.1.1.5.2

Multiply $2$ by $8$ .

$\frac{2 t^{2} e^{2 t} + \ln ({(e^{2 t} + 8)}^{2}) t e^{2 t} + t \cdot \ln ({(e^{2 t} + 8)}^{16})}{e^{2 t} + 8}$

$\frac{2 t^{2} e^{2 t} + \ln ({(e^{2 t} + 8)}^{2}) t e^{2 t} + t \ln ({(e^{2 t} + 8)}^{16})}{e^{2 t} + 8}$

Step 8.1.2

Reorder factors in $2 t^{2} e^{2 t} + \ln ({(e^{2 t} + 8)}^{2}) t e^{2 t} + t \ln ({(e^{2 t} + 8)}^{16})$ .

$\frac{2 t^{2} e^{2 t} + t e^{2 t} \ln ({(e^{2 t} + 8)}^{2}) + t \ln ({(e^{2 t} + 8)}^{16})}{e^{2 t} + 8}$

Step 8.2

Reorder terms.

$\frac{2 e^{2 t} t^{2} + e^{2 t} t \ln ({(e^{2 t} + 8)}^{2}) + t \ln ({(e^{2 t} + 8)}^{16})}{e^{2 t} + 8}$