Calculus Examples

$f (t) = t^{- 4} \sin (t)$

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 $f (t) = t^{- 4}$ and $g (t) = \sin (t)$ .

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

Step 2

The derivative of $\sin (t)$ with respect to $t$ is $\cos (t)$ .

$t^{- 4} \cos (t) + \sin (t) \frac{d}{d t} [t^{- 4}]$

Step 3

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = - 4$ .

$t^{- 4} \cos (t) + \sin (t) (- 4 t^{- 5})$

Step 4

Simplify.

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Step 4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$t^{- 4} \cos (t) + \sin (t) (- 4 \frac{1}{t^{5}})$

Step 4.2

Combine $- 4$ and $\frac{1}{t^{5}}$ .

$t^{- 4} \cos (t) + \sin (t) \frac{- 4}{t^{5}}$

Step 4.3

Move the negative in front of the fraction.

$t^{- 4} \cos (t) + \sin (t) (- \frac{4}{t^{5}})$

Step 5

Simplify.

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Step 5.1

Combine terms.

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Step 5.1.1

Combine $\sin (t)$ and $\frac{4}{t^{5}}$ .

$t^{- 4} \cos (t) - \frac{\sin (t) \cdot 4}{t^{5}}$

Step 5.1.2

Move $4$ to the left of $\sin (t)$ .

$t^{- 4} \cos (t) - \frac{4 \sin (t)}{t^{5}}$

Step 5.2

Simplify each term.

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Step 5.2.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{t^{4}} \cos (t) - \frac{4 \sin (t)}{t^{5}}$

Step 5.2.2

Combine $\frac{1}{t^{4}}$ and $\cos (t)$ .

$\frac{\cos (t)}{t^{4}} - \frac{4 \sin (t)}{t^{5}}$