Calculus Examples

$f (t) = \frac{1}{2} \cdot {(3 t^{2} + t)}^{- 3}$

Step 1

Since $\frac{1}{2}$ is constant with respect to $t$ , the derivative of $\frac{1}{2} \cdot {(3 t^{2} + t)}^{- 3}$ with respect to $t$ is $\frac{1}{2} \frac{d}{d t} [{(3 t^{2} + t)}^{- 3}]$ .

$\frac{1}{2} \frac{d}{d t} [{(3 t^{2} + t)}^{- 3}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{- 3}$ and $g (t) = 3 t^{2} + t$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $3 t^{2} + t$ .

$\frac{1}{2} (\frac{d}{d u} [u^{- 3}] \frac{d}{d t} [3 t^{2} + t])$

Step 2.2

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

$\frac{1}{2} (- 3 u^{- 4} \frac{d}{d t} [3 t^{2} + t])$

Step 2.3

Replace all occurrences of $u$ with $3 t^{2} + t$ .

$\frac{1}{2} (- 3 {(3 t^{2} + t)}^{- 4} \frac{d}{d t} [3 t^{2} + t])$

Step 3

Differentiate.

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Step 3.1

Combine $- 3$ and $\frac{1}{2}$ .

$\frac{- 3}{2} ({(3 t^{2} + t)}^{- 4} \frac{d}{d t} [3 t^{2} + t])$

Step 3.2

Combine fractions.

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Step 3.2.1

Combine ${(3 t^{2} + t)}^{- 4}$ and $\frac{- 3}{2}$ .

$\frac{{(3 t^{2} + t)}^{- 4} \cdot - 3}{2} \frac{d}{d t} [3 t^{2} + t]$

Step 3.2.2

Simplify the expression.

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Step 3.2.2.1

Move $- 3$ to the left of ${(3 t^{2} + t)}^{- 4}$ .

$\frac{- 3 \cdot {(3 t^{2} + t)}^{- 4}}{2} \frac{d}{d t} [3 t^{2} + t]$

Step 3.2.2.2

Move ${(3 t^{2} + t)}^{- 4}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{- 3}{2 {(3 t^{2} + t)}^{4}} \frac{d}{d t} [3 t^{2} + t]$

Step 3.2.2.3

Move the negative in front of the fraction.

$- \frac{3}{2 {(3 t^{2} + t)}^{4}} \frac{d}{d t} [3 t^{2} + t]$

Step 3.3

By the Sum Rule, the derivative of $3 t^{2} + t$ with respect to $t$ is $\frac{d}{d t} [3 t^{2}] + \frac{d}{d t} [t]$ .

$- \frac{3}{2 {(3 t^{2} + t)}^{4}} (\frac{d}{d t} [3 t^{2}] + \frac{d}{d t} [t])$

Step 3.4

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

$- \frac{3}{2 {(3 t^{2} + t)}^{4}} (3 \frac{d}{d t} [t^{2}] + \frac{d}{d t} [t])$

Step 3.5

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

$- \frac{3}{2 {(3 t^{2} + t)}^{4}} (3 (2 t) + \frac{d}{d t} [t])$

Step 3.6

Multiply $2$ by $3$ .

$- \frac{3}{2 {(3 t^{2} + t)}^{4}} (6 t + \frac{d}{d t} [t])$

Step 3.7

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

$- \frac{3}{2 {(3 t^{2} + t)}^{4}} (6 t + 1)$

Step 3.8

Reorder the factors of $- \frac{3}{2 {(3 t^{2} + t)}^{4}} (6 t + 1)$ .

$- (6 t + 1) \frac{3}{2 {(3 t^{2} + t)}^{4}}$