Calculus Examples

$f (x) = t {(\ln (10) t)}^{2}$

Step 1

Apply the product rule to $\ln (10) t$ .

$\frac{d}{d t} [t (\ln^{2} (10) t^{2})]$

Step 2

Reorder factors in $\ln^{2} (10) t^{2}$ .

$\frac{d}{d t} [t (t^{2} \ln^{2} (10))]$

Step 3

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

$\frac{d}{d t} [t^{2} t^{1} \ln^{2} (10)]$

Step 4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{d}{d t} [t^{2 + 1} \ln^{2} (10)]$

Step 5

Add $2$ and $1$ .

$\frac{d}{d t} [t^{3} \ln^{2} (10)]$

Step 6

Since $\ln^{2} (10)$ is constant with respect to $t$ , the derivative of $t^{3} \ln^{2} (10)$ with respect to $t$ is $\ln^{2} (10) \frac{d}{d t} [t^{3}]$ .

$\ln^{2} (10) \frac{d}{d t} [t^{3}]$

Step 7

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 3$ .

$\ln^{2} (10) (3 t^{2})$

Step 8

Reorder.

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Step 8.1

Move $3$ to the left of $\ln^{2} (10)$ .

$3 \cdot \ln^{2} (10) t^{2}$

Step 8.2

Reorder the factors of $3 \ln^{2} (10) t^{2}$ .

$3 t^{2} \ln^{2} (10)$