Calculus Examples

$y = x e^{\frac{1}{x}}$

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 $f (x) = x$ and $g (x) = e^{\frac{1}{x}}$ .

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

Step 2

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 $g (x) = \frac{1}{x}$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $\frac{1}{x}$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $\frac{1}{x}$ .

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

Step 3

Differentiate using the Power Rule.

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Step 3.1

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

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

Step 3.2

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

$x (e^{\frac{1}{x}} (- x^{- 2})) + e^{\frac{1}{x}} \frac{d}{d x} [x]$

Step 4

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

$x^{- 2} x^{1} (e^{\frac{1}{x}} \cdot (- 1)) + e^{\frac{1}{x}} \frac{d}{d x} [x]$

Step 5

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

$x^{- 2 + 1} (e^{\frac{1}{x}} \cdot (- 1)) + e^{\frac{1}{x}} \frac{d}{d x} [x]$

Step 6

Simplify the expression.

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Step 6.1

Add $- 2$ and $1$ .

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

Step 6.2

Move $- 1$ to the left of $e^{\frac{1}{x}}$ .

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

Step 6.3

Rewrite $- 1 e^{\frac{1}{x}}$ as $- e^{\frac{1}{x}}$ .

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

Step 7

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

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

Step 8

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

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

Step 9

Simplify.

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Step 9.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 9.2

Combine $\frac{1}{x}$ and $e^{\frac{1}{x}}$ .

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