Calculus Examples

$y = e^{e^{x}}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (e^{e^{x}})$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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Step 3.1.1

To apply the Chain Rule, set $u$ as $e^{x}$ .

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

Step 3.1.2

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

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

Step 3.1.3

Replace all occurrences of $u$ with $e^{x}$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = e^{e^{x} + x}$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = e^{e^{x} + x}$