Calculus Examples

$e^{x - e} + \ln (x) - \frac{x}{e}$

Step 1

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

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

Step 2

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

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Step 2.1

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 $g (x) = x - e$ .

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Step 2.1.1

To apply the Chain Rule, set $u$ as $x - e$ .

$\frac{d}{d u} [e^{u}] \frac{d}{d x} [x - e] + \frac{d}{d x} [\ln (x)] + \frac{d}{d x} [- \frac{x}{e}]$

Step 2.1.2

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

$e^{u} \frac{d}{d x} [x - e] + \frac{d}{d x} [\ln (x)] + \frac{d}{d x} [- \frac{x}{e}]$

Step 2.1.3

Replace all occurrences of $u$ with $x - e$ .

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

Step 2.2

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

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

Step 2.3

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

$e^{x - e} (1 + \frac{d}{d x} [- e]) + \frac{d}{d x} [\ln (x)] + \frac{d}{d x} [- \frac{x}{e}]$

Step 2.4

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

$e^{x - e} (1 + 0) + \frac{d}{d x} [\ln (x)] + \frac{d}{d x} [- \frac{x}{e}]$

Step 2.5

Add $1$ and $0$ .

$e^{x - e} \cdot 1 + \frac{d}{d x} [\ln (x)] + \frac{d}{d x} [- \frac{x}{e}]$

Step 2.6

Multiply $e^{x - e}$ by $1$ .

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

Step 3

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

$e^{x - e} + \frac{1}{x} + \frac{d}{d x} [- \frac{x}{e}]$

Step 4

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

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Step 4.1

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

$e^{x - e} + \frac{1}{x} - \frac{1}{e} \frac{d}{d x} [x]$

Step 4.2

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

$e^{x - e} + \frac{1}{x} - \frac{1}{e} \cdot 1$

Step 4.3

Multiply $- 1$ by $1$ .

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

Step 5

Simplify.

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Step 5.1

Combine terms.

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Step 5.1.1

To write $e^{x - e}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 5.1.2

Combine the numerators over the common denominator.

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

Step 5.1.3

To write $\frac{e^{x - e} x + 1}{x}$ as a fraction with a common denominator, multiply by $\frac{e}{e}$ .

$\frac{e^{x - e} x + 1}{x} \cdot \frac{e}{e} - \frac{1}{e}$

Step 5.1.4

To write $- \frac{1}{e}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{e^{x - e} x + 1}{x} \cdot \frac{e}{e} - \frac{1}{e} \cdot \frac{x}{x}$

Step 5.1.5

Write each expression with a common denominator of $x e$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.1.5.1

Multiply $\frac{e^{x - e} x + 1}{x}$ by $\frac{e}{e}$ .

$\frac{(e^{x - e} x + 1) e}{x e} - \frac{1}{e} \cdot \frac{x}{x}$

Step 5.1.5.2

Multiply $\frac{1}{e}$ by $\frac{x}{x}$ .

$\frac{(e^{x - e} x + 1) e}{x e} - \frac{x}{e x}$

Step 5.1.5.3

Reorder the factors of $x e$ .

$\frac{(e^{x - e} x + 1) e}{e x} - \frac{x}{e x}$

Step 5.1.6

Combine the numerators over the common denominator.

$\frac{(e^{x - e} x + 1) e - x}{e x}$

Step 5.2

Reorder terms.

$\frac{e (e^{x - e} x + 1) - x}{e x}$