Calculus Examples

$f (x) = 3 x e^{- 9 x^{2}}$

Step 1

Since $3$ is constant with respect to $x$ , the derivative of $3 x e^{- 9 x^{2}}$ with respect to $x$ is $3 \frac{d}{d x} [x e^{- 9 x^{2}}]$ .

$3 \frac{d}{d x} [x e^{- 9 x^{2}}]$

Step 2

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 $g (x) = e^{- 9 x^{2}}$ .

$3 (x \frac{d}{d x} [e^{- 9 x^{2}}] + e^{- 9 x^{2}} \frac{d}{d x} [x])$

Step 3

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 $g (x) = - 9 x^{2}$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $- 9 x^{2}$ .

$3 (x (\frac{d}{d u} [e^{u}] \frac{d}{d x} [- 9 x^{2}]) + e^{- 9 x^{2}} \frac{d}{d x} [x])$

Step 3.2

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

$3 (x (e^{u} \frac{d}{d x} [- 9 x^{2}]) + e^{- 9 x^{2}} \frac{d}{d x} [x])$

Step 3.3

Replace all occurrences of $u$ with $- 9 x^{2}$ .

$3 (x (e^{- 9 x^{2}} \frac{d}{d x} [- 9 x^{2}]) + e^{- 9 x^{2}} \frac{d}{d x} [x])$

Step 4

Differentiate.

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Step 4.1

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9 x^{2}$ with respect to $x$ is $- 9 \frac{d}{d x} [x^{2}]$ .

$3 (x (e^{- 9 x^{2}} (- 9 \frac{d}{d x} [x^{2}])) + e^{- 9 x^{2}} \frac{d}{d x} [x])$

Step 4.2

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

$3 (x (e^{- 9 x^{2}} (- 9 (2 x))) + e^{- 9 x^{2}} \frac{d}{d x} [x])$

Step 4.3

Multiply $2$ by $- 9$ .

$3 (x (e^{- 9 x^{2}} (- 18 x)) + e^{- 9 x^{2}} \frac{d}{d x} [x])$

Step 5

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

$3 (x^{1} x (e^{- 9 x^{2}} \cdot (- 18)) + e^{- 9 x^{2}} \frac{d}{d x} [x])$

Step 6

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

$3 (x^{1} x^{1} (e^{- 9 x^{2}} \cdot (- 18)) + e^{- 9 x^{2}} \frac{d}{d x} [x])$

Step 7

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

$3 (x^{1 + 1} (e^{- 9 x^{2}} \cdot (- 18)) + e^{- 9 x^{2}} \frac{d}{d x} [x])$

Step 8

Simplify the expression.

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Step 8.1

Add $1$ and $1$ .

$3 (x^{2} (e^{- 9 x^{2}} \cdot (- 18)) + e^{- 9 x^{2}} \frac{d}{d x} [x])$

Step 8.2

Move $- 18$ to the left of $e^{- 9 x^{2}}$ .

$3 (x^{2} (- 18 \cdot e^{- 9 x^{2}}) + e^{- 9 x^{2}} \frac{d}{d x} [x])$

Step 9

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

$3 (x^{2} (- 18 e^{- 9 x^{2}}) + e^{- 9 x^{2}} \cdot 1)$

Step 10

Multiply $e^{- 9 x^{2}}$ by $1$ .

$3 (x^{2} (- 18 e^{- 9 x^{2}}) + e^{- 9 x^{2}})$

Step 11

Simplify.

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Step 11.1

Apply the distributive property.

$3 (x^{2} (- 18 e^{- 9 x^{2}})) + 3 e^{- 9 x^{2}}$

Step 11.2

Multiply $- 18$ by $3$ .

$- 54 x^{2} e^{- 9 x^{2}} + 3 e^{- 9 x^{2}}$

Step 11.3

Reorder terms.

$- 54 e^{- 9 x^{2}} x^{2} + 3 e^{- 9 x^{2}}$

Step 11.4

Reorder factors in $- 54 e^{- 9 x^{2}} x^{2} + 3 e^{- 9 x^{2}}$ .

$- 54 x^{2} e^{- 9 x^{2}} + 3 e^{- 9 x^{2}}$