Calculus Examples

$y = x^{2} e^{- x^{2}}$

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 $f (x) = x^{2}$ and $g (x) = e^{- x^{2}}$ .

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

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 $g (x) = - x^{2}$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $- x^{2}$ .

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

Step 3

Differentiate.

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Step 3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

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

Step 3.2

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

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

Step 3.3

Multiply $2$ by $- 1$ .

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

Step 4

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 4.1

Move $x$ .

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

Step 4.2

Multiply $x$ by $x^{2}$ .

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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.3

Add $1$ and $2$ .

$x^{3} (e^{- x^{2}} \cdot (- 2)) + e^{- x^{2}} \frac{d}{d x} [x^{2}]$

Step 5

Move $- 2$ to the left of $e^{- x^{2}}$ .

$x^{3} (- 2 \cdot e^{- x^{2}}) + e^{- x^{2}} \frac{d}{d x} [x^{2}]$

Step 6

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

$x^{3} (- 2 e^{- x^{2}}) + e^{- x^{2}} (2 x)$

Step 7

Simplify.

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Step 7.1

Reorder terms.

$- 2 e^{- x^{2}} x^{3} + 2 e^{- x^{2}} x$

Step 7.2

Reorder factors in $- 2 e^{- x^{2}} x^{3} + 2 e^{- x^{2}} x$ .

$- 2 x^{3} e^{- x^{2}} + 2 x e^{- x^{2}}$