Calculus Examples

$y = e^{5 e^{x}}$

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 $g (x) = 5 e^{x}$ .

Tap for more steps...

Step 1.1

To apply the Chain Rule, set $u$ as $5 e^{x}$ .

$\frac{d}{d u} [e^{u}] \frac{d}{d x} [5 e^{x}]$

Step 1.2

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

$e^{u} \frac{d}{d x} [5 e^{x}]$

Step 1.3

Replace all occurrences of $u$ with $5 e^{x}$ .

$e^{5 e^{x}} \frac{d}{d x} [5 e^{x}]$

Step 2

Since $5$ is constant with respect to $x$ , the derivative of $5 e^{x}$ with respect to $x$ is $5 \frac{d}{d x} [e^{x}]$ .

$e^{5 e^{x}} (5 \frac{d}{d x} [e^{x}])$

Step 3

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

$e^{5 e^{x}} (5 e^{x})$

Step 4

Multiply $e^{5 e^{x}}$ by $e^{x}$ by adding the exponents.

Tap for more steps...

Step 4.1

Move $e^{x}$ .

$e^{x} e^{5 e^{x}} \cdot 5$

Step 4.2

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

$e^{x + 5 e^{x}} \cdot 5$

Step 5

Move $5$ to the left of $e^{x + 5 e^{x}}$ .

$5 e^{x + 5 e^{x}}$