Calculus Examples

$C e^{- x^{3}}$

Step 1

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

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

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 $g (x) = - x^{3}$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $- x^{3}$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $- x^{3}$ .

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

Step 3

Differentiate.

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Step 3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{3}$ with respect to $x$ is $- \frac{d}{d x} [x^{3}]$ .

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

Step 3.2

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

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

Step 3.3

Multiply $3$ by $- 1$ .

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

Step 4

Simplify.

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Step 4.1

Reorder the factors of $C e^{- x^{3}} (- 3 x^{2})$ .

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

Step 4.2

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

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