Calculus Examples

$e^{e^{x^{2}}}$

Step 1

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 $g (x) = e^{x^{2}}$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $e^{x^{2}}$ .

$\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [e^{x^{2}}]$

Step 1.2

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

$e^{u_{1}} \frac{d}{d x} [e^{x^{2}}]$

Step 1.3

Replace all occurrences of $u_{1}$ with $e^{x^{2}}$ .

$e^{e^{x^{2}}} \frac{d}{d x} [e^{x^{2}}]$

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 $g (x) = x^{2}$ .

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Step 2.1

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

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

Step 2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{2}} [a^{u_{2}}]$ is $a^{u_{2}} \ln (a)$ where $a$ = $e$ .

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

Step 2.3

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

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

Step 3

Multiply $e^{e^{x^{2}}}$ by $e^{x^{2}}$ by adding the exponents.

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Step 3.1

Move $e^{x^{2}}$ .

$e^{x^{2}} e^{e^{x^{2}}} \frac{d}{d x} [x^{2}]$

Step 3.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$e^{x^{2} + e^{x^{2}}} \frac{d}{d x} [x^{2}]$

Step 4

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

$e^{x^{2} + e^{x^{2}}} (2 x)$

Step 5

Simplify.

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Step 5.1

Reorder the factors of $e^{x^{2} + e^{x^{2}}} (2 x)$ .

$2 e^{x^{2} + e^{x^{2}}} x$

Step 5.2

Reorder factors in $2 e^{x^{2} + e^{x^{2}}} x$ .

$2 x e^{x^{2} + e^{x^{2}}}$