Calculus Examples

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

Step 1

Differentiate.

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Step 1.1

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

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

Step 1.2

Since $e$ is constant with respect to $x$ , the derivative of $e$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- e^{x}] + \frac{d}{d x} [x^{e}]$

Step 2

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

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Step 2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- e^{x}$ with respect to $x$ is $- \frac{d}{d x} [e^{x}]$ .

$0 - \frac{d}{d x} [e^{x}] + \frac{d}{d x} [x^{e}]$

Step 2.2

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

$0 - e^{x} + \frac{d}{d x} [x^{e}]$

Step 3

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

$0 - e^{x} + e x^{e - 1}$

Step 4

Simplify.

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Step 4.1

Subtract $e^{x}$ from $0$ .

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

Step 4.2

Reorder terms.

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