Calculus Examples

$y = 8 e^{\tan (x^{2})}$

Step 1

Since $8$ is constant with respect to $x$ , the derivative of $8 e^{\tan (x^{2})}$ with respect to $x$ is $8 \frac{d}{d x} [e^{\tan (x^{2})}]$ .

$8 \frac{d}{d x} [e^{\tan (x^{2})}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = \tan (x^{2})$ .

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Step 2.1

To apply the Chain Rule, set $u_{1}$ as $\tan (x^{2})$ .

$8 (\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [\tan (x^{2})])$

Step 2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$8 (e^{u_{1}} \frac{d}{d x} [\tan (x^{2})])$

Step 2.3

Replace all occurrences of $u_{1}$ with $\tan (x^{2})$ .

$8 (e^{\tan (x^{2})} \frac{d}{d x} [\tan (x^{2})])$

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \tan (x)$ and $g (x) = x^{2}$ .

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Step 3.1

To apply the Chain Rule, set $u_{2}$ as $x^{2}$ .

$8 e^{\tan (x^{2})} (\frac{d}{d u_{2}} [\tan (u_{2})] \frac{d}{d x} [x^{2}])$

Step 3.2

The derivative of $\tan (u_{2})$ with respect to $u_{2}$ is $\sec^{2} (u_{2})$ .

$8 e^{\tan (x^{2})} (\sec^{2} (u_{2}) \frac{d}{d x} [x^{2}])$

Step 3.3

Replace all occurrences of $u_{2}$ with $x^{2}$ .

$8 e^{\tan (x^{2})} (\sec^{2} (x^{2}) \frac{d}{d x} [x^{2}])$

Step 4

Differentiate using the Power Rule.

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Step 4.1

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

$8 e^{\tan (x^{2})} \sec^{2} (x^{2}) (2 x)$

Step 4.2

Multiply $2$ by $8$ .

$16 e^{\tan (x^{2})} \sec^{2} (x^{2}) x$

Step 5

Simplify.

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Step 5.1

Reorder the factors of $16 e^{\tan (x^{2})} \sec^{2} (x^{2}) x$ .

$16 e^{\tan (x^{2})} x \sec^{2} (x^{2})$

Step 5.2

Reorder factors in $16 e^{\tan (x^{2})} x \sec^{2} (x^{2})$ .

$16 x e^{\tan (x^{2})} \sec^{2} (x^{2})$