Calculus Examples

$\sin^{2} (e^{\sin^{2} (t)})$

Step 1

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

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $\sin (e^{\sin^{2} (t)})$ .

$\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d t} [\sin (e^{\sin^{2} (t)})]$

Step 1.2

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

$2 u_{1} \frac{d}{d t} [\sin (e^{\sin^{2} (t)})]$

Step 1.3

Replace all occurrences of $u_{1}$ with $\sin (e^{\sin^{2} (t)})$ .

$2 \sin (e^{\sin^{2} (t)}) \frac{d}{d t} [\sin (e^{\sin^{2} (t)})]$

Step 2

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

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Step 2.1

To apply the Chain Rule, set $u_{2}$ as $e^{\sin^{2} (t)}$ .

$2 \sin (e^{\sin^{2} (t)}) (\frac{d}{d u_{2}} [\sin (u_{2})] \frac{d}{d t} [e^{\sin^{2} (t)}])$

Step 2.2

The derivative of $\sin (u_{2})$ with respect to $u_{2}$ is $\cos (u_{2})$ .

$2 \sin (e^{\sin^{2} (t)}) (\cos (u_{2}) \frac{d}{d t} [e^{\sin^{2} (t)}])$

Step 2.3

Replace all occurrences of $u_{2}$ with $e^{\sin^{2} (t)}$ .

$2 \sin (e^{\sin^{2} (t)}) (\cos (e^{\sin^{2} (t)}) \frac{d}{d t} [e^{\sin^{2} (t)}])$

Step 3

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

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Step 3.1

To apply the Chain Rule, set $u_{3}$ as $\sin^{2} (t)$ .

$2 \sin (e^{\sin^{2} (t)}) \cos (e^{\sin^{2} (t)}) (\frac{d}{d u_{3}} [e^{u_{3}}] \frac{d}{d t} [\sin^{2} (t)])$

Step 3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{3}} [a^{u_{3}}]$ is $a^{u_{3}} \ln (a)$ where $a$ = $e$ .

$2 \sin (e^{\sin^{2} (t)}) \cos (e^{\sin^{2} (t)}) (e^{u_{3}} \frac{d}{d t} [\sin^{2} (t)])$

Step 3.3

Replace all occurrences of $u_{3}$ with $\sin^{2} (t)$ .

$2 \sin (e^{\sin^{2} (t)}) \cos (e^{\sin^{2} (t)}) (e^{\sin^{2} (t)} \frac{d}{d t} [\sin^{2} (t)])$

Step 4

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

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Step 4.1

To apply the Chain Rule, set $u_{4}$ as $\sin (t)$ .

$2 \sin (e^{\sin^{2} (t)}) \cos (e^{\sin^{2} (t)}) e^{\sin^{2} (t)} (\frac{d}{d u_{4}} [{u_{4}}^{2}] \frac{d}{d t} [\sin (t)])$

Step 4.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{4}} [{u_{4}}^{n}]$ is $n {u_{4}}^{n - 1}$ where $n = 2$ .

$2 \sin (e^{\sin^{2} (t)}) \cos (e^{\sin^{2} (t)}) e^{\sin^{2} (t)} (2 u_{4} \frac{d}{d t} [\sin (t)])$

Step 4.3

Replace all occurrences of $u_{4}$ with $\sin (t)$ .

$2 \sin (e^{\sin^{2} (t)}) \cos (e^{\sin^{2} (t)}) e^{\sin^{2} (t)} (2 \sin (t) \frac{d}{d t} [\sin (t)])$

Step 5

Multiply $2$ by $2$ .

$4 \sin (e^{\sin^{2} (t)}) \cos (e^{\sin^{2} (t)}) e^{\sin^{2} (t)} (\sin (t) \frac{d}{d t} [\sin (t)])$

Step 6

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

$4 \sin (e^{\sin^{2} (t)}) \cos (e^{\sin^{2} (t)}) e^{\sin^{2} (t)} \sin (t) \cos (t)$

Step 7

Reorder the factors of $4 \sin (e^{\sin^{2} (t)}) \cos (e^{\sin^{2} (t)}) e^{\sin^{2} (t)} \sin (t) \cos (t)$ .

$4 e^{\sin^{2} (t)} \cos (e^{\sin^{2} (t)}) \cos (t) \sin (e^{\sin^{2} (t)}) \sin (t)$