Calculus Examples

$f (x) = 2 e^{e^{2 x}}$

Step 1

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

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

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

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Step 2.1

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

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

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 $a$ = $e$ .

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

Step 2.3

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

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

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

Step 5

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

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

Step 6

Multiply $2$ by $2$ .

$4 e^{2 x + e^{2 x}} \frac{d}{d x} [x]$

Step 7

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

$4 e^{2 x + e^{2 x}} \cdot 1$

Step 8

Multiply $4$ by $1$ .

$4 e^{2 x + e^{2 x}}$