Calculus Examples

$f (x) = 3 e^{- x^{4}}$

Step 1

Find the first derivative.

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Step 1.1

Since $3$ is constant with respect to $x$ , the derivative of $3 e^{- x^{4}}$ with respect to $x$ is $3 \frac{d}{d x} [e^{- x^{4}}]$ .

$3 \frac{d}{d x} [e^{- x^{4}}]$

Step 1.2

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 $g (x) = - x^{4}$ .

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Step 1.2.1

To apply the Chain Rule, set $u$ as $- x^{4}$ .

$3 (\frac{d}{d u} [e^{u}] \frac{d}{d x} [- x^{4}])$

Step 1.2.2

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

$3 (e^{u} \frac{d}{d x} [- x^{4}])$

Step 1.2.3

Replace all occurrences of $u$ with $- x^{4}$ .

$3 (e^{- x^{4}} \frac{d}{d x} [- x^{4}])$

Step 1.3

Differentiate.

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Step 1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{4}$ with respect to $x$ is $- \frac{d}{d x} [x^{4}]$ .

$3 e^{- x^{4}} (- \frac{d}{d x} [x^{4}])$

Step 1.3.2

Multiply $- 1$ by $3$ .

$- 3 e^{- x^{4}} \frac{d}{d x} [x^{4}]$

Step 1.3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$- 3 e^{- x^{4}} (4 x^{3})$

Step 1.3.4

Simplify the expression.

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Step 1.3.4.1

Multiply $4$ by $- 3$ .

$- 12 e^{- x^{4}} x^{3}$

Step 1.3.4.2

Reorder factors in $- 12 e^{- x^{4}} x^{3}$ .

$f' (x) = - 12 x^{3} e^{- x^{4}}$

Step 2

Find the second derivative.

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Step 2.1

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12 x^{3} e^{- x^{4}}$ with respect to $x$ is $- 12 \frac{d}{d x} [x^{3} e^{- x^{4}}]$ .

$- 12 \frac{d}{d x} [x^{3} e^{- x^{4}}]$

Step 2.2

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 $f (x) = x^{3}$ and $g (x) = e^{- x^{4}}$ .

$- 12 (x^{3} \frac{d}{d x} [e^{- x^{4}}] + e^{- x^{4}} \frac{d}{d x} [x^{3}])$

Step 2.3

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 $g (x) = - x^{4}$ .

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Step 2.3.1

To apply the Chain Rule, set $u$ as $- x^{4}$ .

$- 12 (x^{3} (\frac{d}{d u} [e^{u}] \frac{d}{d x} [- x^{4}]) + e^{- x^{4}} \frac{d}{d x} [x^{3}])$

Step 2.3.2

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

$- 12 (x^{3} (e^{u} \frac{d}{d x} [- x^{4}]) + e^{- x^{4}} \frac{d}{d x} [x^{3}])$

Step 2.3.3

Replace all occurrences of $u$ with $- x^{4}$ .

$- 12 (x^{3} (e^{- x^{4}} \frac{d}{d x} [- x^{4}]) + e^{- x^{4}} \frac{d}{d x} [x^{3}])$

Step 2.4

Differentiate.

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Step 2.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{4}$ with respect to $x$ is $- \frac{d}{d x} [x^{4}]$ .

$- 12 (x^{3} (e^{- x^{4}} (- \frac{d}{d x} [x^{4}])) + e^{- x^{4}} \frac{d}{d x} [x^{3}])$

Step 2.4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$- 12 (x^{3} (e^{- x^{4}} (- (4 x^{3}))) + e^{- x^{4}} \frac{d}{d x} [x^{3}])$

Step 2.4.3

Multiply $4$ by $- 1$ .

$- 12 (x^{3} (e^{- x^{4}} (- 4 x^{3})) + e^{- x^{4}} \frac{d}{d x} [x^{3}])$

Step 2.5

Multiply $x^{3}$ by $x^{3}$ by adding the exponents.

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Step 2.5.1

Move $x^{3}$ .

$- 12 (x^{3} x^{3} (e^{- x^{4}} \cdot (- 4)) + e^{- x^{4}} \frac{d}{d x} [x^{3}])$

Step 2.5.2

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

$- 12 (x^{3 + 3} (e^{- x^{4}} \cdot (- 4)) + e^{- x^{4}} \frac{d}{d x} [x^{3}])$

Step 2.5.3

Add $3$ and $3$ .

$- 12 (x^{6} (e^{- x^{4}} \cdot (- 4)) + e^{- x^{4}} \frac{d}{d x} [x^{3}])$

Step 2.6

Move $- 4$ to the left of $e^{- x^{4}}$ .

$- 12 (x^{6} (- 4 \cdot e^{- x^{4}}) + e^{- x^{4}} \frac{d}{d x} [x^{3}])$

Step 2.7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$- 12 (x^{6} (- 4 e^{- x^{4}}) + e^{- x^{4}} (3 x^{2}))$

Step 2.8

Simplify.

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Step 2.8.1

Apply the distributive property.

$- 12 (x^{6} (- 4 e^{- x^{4}})) - 12 (e^{- x^{4}} (3 x^{2}))$

Step 2.8.2

Combine terms.

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Step 2.8.2.1

Multiply $- 4$ by $- 12$ .

$48 (x^{6} (e^{- x^{4}})) - 12 (e^{- x^{4}} (3 x^{2}))$

Step 2.8.2.2

Multiply $3$ by $- 12$ .

$48 x^{6} e^{- x^{4}} - 36 e^{- x^{4}} x^{2}$

Step 2.8.3

Reorder terms.

$48 e^{- x^{4}} x^{6} - 36 e^{- x^{4}} x^{2}$

Step 2.8.4

Reorder factors in $48 e^{- x^{4}} x^{6} - 36 e^{- x^{4}} x^{2}$ .

$f'' (x) = 48 x^{6} e^{- x^{4}} - 36 x^{2} e^{- x^{4}}$

Step 3

The second derivative of $f (x)$ with respect to $x$ is $48 x^{6} e^{- x^{4}} - 36 x^{2} e^{- x^{4}}$ .

$48 x^{6} e^{- x^{4}} - 36 x^{2} e^{- x^{4}}$