Calculus Examples

$y = e^{t} \sin (t)$

Find the first derivative.

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

$f' (t) = e^{t} \frac{d}{d t} (\sin (t)) + \sin (t) \frac{d}{d t} (e^{t})$

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

$f' (t) = e^{t} \cos (t) + \sin (t) \frac{d}{d t} (e^{t})$

Differentiate using the Exponential Rule which states that $\frac{d}{d t} [a^{t}]$ is $a^{t} \ln (a)$ where $a$ = $e$ .

$f' (t) = e^{t} \cos (t) + \sin (t) e^{t}$

Reorder terms.

$f' (t) = e^{t} \cos (t) + e^{t} \sin (t)$

Find the second derivative.

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By the Sum Rule, the derivative of $e^{t} \cos (t) + e^{t} \sin (t)$ with respect to $t$ is $\frac{d}{d t} [e^{t} \cos (t)] + \frac{d}{d t} [e^{t} \sin (t)]$ .

$f'' (t) = \frac{d}{d t} (e^{t} \cos (t)) + \frac{d}{d t} (e^{t} \sin (t))$

Evaluate $\frac{d}{d t} [e^{t} \cos (t)]$ .

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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 $f (t) = e^{t}$ and $g (t) = \cos (t)$ .

$f'' (t) = e^{t} \frac{d}{d t} (\cos (t)) + \cos (t) \frac{d}{d t} (e^{t}) + \frac{d}{d t} (e^{t} \sin (t))$

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

$f'' (t) = e^{t} (- \sin (t)) + \cos (t) \frac{d}{d t} (e^{t}) + \frac{d}{d t} (e^{t} \sin (t))$

Differentiate using the Exponential Rule which states that $\frac{d}{d t} [a^{t}]$ is $a^{t} \ln (a)$ where $a$ = $e$ .

$f'' (t) = e^{t} (- \sin (t)) + \cos (t) e^{t} + \frac{d}{d t} (e^{t} \sin (t))$

Evaluate $\frac{d}{d t} [e^{t} \sin (t)]$ .

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

$f'' (t) = e^{t} (- \sin (t)) + \cos (t) e^{t} + e^{t} \frac{d}{d t} (\sin (t)) + \sin (t) \frac{d}{d t} (e^{t})$

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

$f'' (t) = e^{t} (- \sin (t)) + \cos (t) e^{t} + e^{t} \cos (t) + \sin (t) \frac{d}{d t} (e^{t})$

Differentiate using the Exponential Rule which states that $\frac{d}{d t} [a^{t}]$ is $a^{t} \ln (a)$ where $a$ = $e$ .

$f'' (t) = e^{t} (- \sin (t)) + \cos (t) e^{t} + e^{t} \cos (t) + \sin (t) e^{t}$

Combine terms.

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Add $\cos (t) e^{t}$ and $e^{t} \cos (t)$ .

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Reorder $\cos (t)$ and $e^{t}$ .

$f'' (t) = e^{t} (- \sin (t)) + e^{t} \cos (t) + e^{t} \cos (t) + \sin (t) e^{t}$

Add $e^{t} \cos (t)$ and $e^{t} \cos (t)$ .

$f'' (t) = e^{t} (- \sin (t)) + 2 e^{t} \cos (t) + \sin (t) e^{t}$

Reorder $\sin (t)$ and $e^{t}$ .

$f'' (t) = 2 e^{t} \cos (t) - 1 \cdot (e^{t} \sin (t)) + e^{t} \sin (t)$

Rewrite $- 1 e^{t}$ as $- e^{t}$ .

$f'' (t) = 2 e^{t} \cos (t) - e^{t} \sin (t) + e^{t} \sin (t)$

Add $- e^{t} \sin (t)$ and $e^{t} \sin (t)$ .

$f'' (t) = 2 e^{t} \cos (t) + 0$

Add $2 e^{t} \cos (t)$ and $0$ .

$f'' (t) = 2 e^{t} \cos (t)$