Calculus Examples

$18 e^{x} - e x^{e}$

Step 1

By the Sum Rule, the derivative of $18 e^{x} - e x^{e}$ with respect to $x$ is $\frac{d}{d x} [18 e^{x}] + \frac{d}{d x} [- e x^{e}]$ .

$\frac{d}{d x} [18 e^{x}] + \frac{d}{d x} [- e x^{e}]$

Step 2

Evaluate $\frac{d}{d x} [18 e^{x}]$ .

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Step 2.1

Since $18$ is constant with respect to $x$ , the derivative of $18 e^{x}$ with respect to $x$ is $18 \frac{d}{d x} [e^{x}]$ .

$18 \frac{d}{d x} [e^{x}] + \frac{d}{d x} [- e x^{e}]$

Step 2.2

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

$18 e^{x} + \frac{d}{d x} [- e x^{e}]$

Step 3

Evaluate $\frac{d}{d x} [- e x^{e}]$ .

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Step 3.1

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

$18 e^{x} - e \frac{d}{d x} [x^{e}]$

Step 3.2

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

$18 e^{x} - e (e x^{e - 1})$

Step 3.3

Raise $e$ to the power of $1$ .

$18 e^{x} - (e^{1} e) x^{e - 1}$

Step 3.4

Raise $e$ to the power of $1$ .

$18 e^{x} - (e^{1} e^{1}) x^{e - 1}$

Step 3.5

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

$18 e^{x} - e^{1 + 1} x^{e - 1}$

Step 3.6

Add $1$ and $1$ .

$18 e^{x} - e^{2} x^{e - 1}$

Step 4

Reorder terms.

$- e^{2} x^{e - 1} + 18 e^{x}$