Calculus Examples

$s = t^{7} \ln (t)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d t} (s) = \frac{d}{d t} (t^{7} \ln (t))$

Step 2

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

$s'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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 $f (t) = t^{7}$ and $g (t) = \ln (t)$ .

$t^{7} \frac{d}{d t} [\ln (t)] + \ln (t) \frac{d}{d t} [t^{7}]$

Step 3.2

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

$t^{7} \frac{1}{t} + \ln (t) \frac{d}{d t} [t^{7}]$

Step 3.3

Differentiate using the Power Rule.

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Step 3.3.1

Combine $t^{7}$ and $\frac{1}{t}$ .

$\frac{t^{7}}{t} + \ln (t) \frac{d}{d t} [t^{7}]$

Step 3.3.2

Cancel the common factor of $t^{7}$ and $t$ .

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Step 3.3.2.1

Factor $t$ out of $t^{7}$ .

$\frac{t \cdot t^{6}}{t} + \ln (t) \frac{d}{d t} [t^{7}]$

Step 3.3.2.2

Cancel the common factors.

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Step 3.3.2.2.1

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

$\frac{t \cdot t^{6}}{t^{1}} + \ln (t) \frac{d}{d t} [t^{7}]$

Step 3.3.2.2.2

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

$\frac{t \cdot t^{6}}{t \cdot 1} + \ln (t) \frac{d}{d t} [t^{7}]$

Step 3.3.2.2.3

Cancel the common factor.

$\frac{t \cdot t^{6}}{t \cdot 1} + \ln (t) \frac{d}{d t} [t^{7}]$

Step 3.3.2.2.4

Rewrite the expression.

$\frac{t^{6}}{1} + \ln (t) \frac{d}{d t} [t^{7}]$

Step 3.3.2.2.5

Divide $t^{6}$ by $1$ .

$t^{6} + \ln (t) \frac{d}{d t} [t^{7}]$

Step 3.3.3

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 7$ .

$t^{6} + \ln (t) (7 t^{6})$

Step 3.3.4

Reorder terms.

$t^{6} + 7 t^{6} \ln (t)$

Step 4

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

$s' = t^{6} + 7 t^{6} \ln (t)$

Step 5

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

$\frac{d s}{d t} = t^{6} + 7 t^{6} \ln (t)$