Calculus Examples

$y = e^{\frac{1}{x^{2}}}$

Step 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 $g (x) = \frac{1}{x^{2}}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $\frac{1}{x^{2}}$ .

$\frac{d}{d u} [e^{u}] \frac{d}{d x} [\frac{1}{x^{2}}]$

Step 1.2

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

$e^{u} \frac{d}{d x} [\frac{1}{x^{2}}]$

Step 1.3

Replace all occurrences of $u$ with $\frac{1}{x^{2}}$ .

$e^{\frac{1}{x^{2}}} \frac{d}{d x} [\frac{1}{x^{2}}]$

Step 2

Differentiate using the Power Rule.

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Step 2.1

Rewrite $\frac{1}{x^{2}}$ as ${(x^{2})}^{- 1}$ .

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

Step 2.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 2.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.2.2

Multiply $2$ by $- 1$ .

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

Step 2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 2$ .

$e^{\frac{1}{x^{2}}} (- 2 x^{- 3})$

Step 3

Simplify.

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Step 3.1

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

$e^{\frac{1}{x^{2}}} (- 2 \frac{1}{x^{3}})$

Step 3.2

Combine terms.

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Step 3.2.1

Combine $- 2$ and $\frac{1}{x^{3}}$ .

$e^{\frac{1}{x^{2}}} \frac{- 2}{x^{3}}$

Step 3.2.2

Move the negative in front of the fraction.

$e^{\frac{1}{x^{2}}} (- \frac{2}{x^{3}})$

Step 3.2.3

Combine $e^{\frac{1}{x^{2}}}$ and $\frac{2}{x^{3}}$ .

$- \frac{e^{\frac{1}{x^{2}}} \cdot 2}{x^{3}}$

Step 3.2.4

Move $2$ to the left of $e^{\frac{1}{x^{2}}}$ .

$- \frac{2 e^{\frac{1}{x^{2}}}}{x^{3}}$