Calculus Examples

$P = 9 t^{5} \ln (8 t)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d t} (P) = \frac{d}{d t} (9 t^{5} \ln (8 t))$

Step 2

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

$P'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Since $9$ is constant with respect to $t$ , the derivative of $9 t^{5} \ln (8 t)$ with respect to $t$ is $9 \frac{d}{d t} [t^{5} \ln (8 t)]$ .

$9 \frac{d}{d t} [t^{5} \ln (8 t)]$

Step 3.2

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 $f (t) = t^{5}$ and $g (t) = \ln (8 t)$ .

$9 (t^{5} \frac{d}{d t} [\ln (8 t)] + \ln (8 t) \frac{d}{d t} [t^{5}])$

Step 3.3

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 $g (t) = 8 t$ .

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Step 3.3.1

To apply the Chain Rule, set $u$ as $8 t$ .

$9 (t^{5} (\frac{d}{d u} [\ln (u)] \frac{d}{d t} [8 t]) + \ln (8 t) \frac{d}{d t} [t^{5}])$

Step 3.3.2

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

$9 (t^{5} (\frac{1}{u} \frac{d}{d t} [8 t]) + \ln (8 t) \frac{d}{d t} [t^{5}])$

Step 3.3.3

Replace all occurrences of $u$ with $8 t$ .

$9 (t^{5} (\frac{1}{8 t} \frac{d}{d t} [8 t]) + \ln (8 t) \frac{d}{d t} [t^{5}])$

Step 3.4

Differentiate.

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Step 3.4.1

Combine $\frac{1}{8 t}$ and $t^{5}$ .

$9 (\frac{t^{5}}{8 t} \frac{d}{d t} [8 t] + \ln (8 t) \frac{d}{d t} [t^{5}])$

Step 3.4.2

Cancel the common factor of $t^{5}$ and $t$ .

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Step 3.4.2.1

Factor $t$ out of $t^{5}$ .

$9 (\frac{t \cdot t^{4}}{8 t} \frac{d}{d t} [8 t] + \ln (8 t) \frac{d}{d t} [t^{5}])$

Step 3.4.2.2

Cancel the common factors.

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Step 3.4.2.2.1

Factor $t$ out of $8 t$ .

$9 (\frac{t \cdot t^{4}}{t \cdot 8} \frac{d}{d t} [8 t] + \ln (8 t) \frac{d}{d t} [t^{5}])$

Step 3.4.2.2.2

Cancel the common factor.

$9 (\frac{t \cdot t^{4}}{t \cdot 8} \frac{d}{d t} [8 t] + \ln (8 t) \frac{d}{d t} [t^{5}])$

Step 3.4.2.2.3

Rewrite the expression.

$9 (\frac{t^{4}}{8} \frac{d}{d t} [8 t] + \ln (8 t) \frac{d}{d t} [t^{5}])$

Step 3.4.3

Since $8$ is constant with respect to $t$ , the derivative of $8 t$ with respect to $t$ is $8 \frac{d}{d t} [t]$ .

$9 (\frac{t^{4}}{8} (8 \frac{d}{d t} [t]) + \ln (8 t) \frac{d}{d t} [t^{5}])$

Step 3.4.4

Simplify terms.

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Step 3.4.4.1

Combine $8$ and $\frac{t^{4}}{8}$ .

$9 (\frac{8 t^{4}}{8} \frac{d}{d t} [t] + \ln (8 t) \frac{d}{d t} [t^{5}])$

Step 3.4.4.2

Cancel the common factor of $8$ .

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Step 3.4.4.2.1

Cancel the common factor.

$9 (\frac{8 t^{4}}{8} \frac{d}{d t} [t] + \ln (8 t) \frac{d}{d t} [t^{5}])$

Step 3.4.4.2.2

Divide $t^{4}$ by $1$ .

$9 (t^{4} \frac{d}{d t} [t] + \ln (8 t) \frac{d}{d t} [t^{5}])$

Step 3.4.5

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

$9 (t^{4} \cdot 1 + \ln (8 t) \frac{d}{d t} [t^{5}])$

Step 3.4.6

Multiply $t^{4}$ by $1$ .

$9 (t^{4} + \ln (8 t) \frac{d}{d t} [t^{5}])$

Step 3.4.7

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 5$ .

$9 (t^{4} + \ln (8 t) (5 t^{4}))$

Step 3.5

Simplify.

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Step 3.5.1

Apply the distributive property.

$9 t^{4} + 9 (\ln (8 t) (5 t^{4}))$

Step 3.5.2

Multiply $5$ by $9$ .

$9 t^{4} + 45 \ln (8 t) t^{4}$

Step 3.5.3

Reorder terms.

$9 t^{4} + 45 t^{4} \ln (8 t)$

Step 4

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

$P' = 9 t^{4} + 45 t^{4} \ln (8 t)$

Step 5

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

$\frac{d P}{d t} = 9 t^{4} + 45 t^{4} \ln (8 t)$