Calculus Examples

$\frac{d}{d x} (e^{x} \cdot e^{x})$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = e^{x}$ and $g (x) = e^{x}$ .

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

Step 2

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

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

Step 3

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

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

Step 4

Combine terms.

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

Add $x$ and $x$ .

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

Add $x$ and $x$ .

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

Step 4.3

Add $e^{2 x}$ and $e^{2 x}$ .

$2 e^{2 x}$