Calculus Examples

$f (s) = (s^{2} + 1) e^{- s^{2}}$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d s} [f (s) g (s)]$ is $f (s) \frac{d}{d s} [g (s)] + g (s) \frac{d}{d s} [f (s)]$ where $f (s) = s^{2} + 1$ and $g (s) = e^{- s^{2}}$ .

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d s} [f (g (s))]$ is $f' (g (s)) g' (s)$ where $f (s) = e^{s}$ and $g (s) = - s^{2}$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Multiply $2$ by $- 1$ .

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

Step 3.4

By the Sum Rule, the derivative of $s^{2} + 1$ with respect to $s$ is $\frac{d}{d s} [s^{2}] + \frac{d}{d s} [1]$ .

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

Step 3.5

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

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

Step 3.6

Since $1$ is constant with respect to $s$ , the derivative of $1$ with respect to $s$ is $0$ .

$(s^{2} + 1) e^{- s^{2}} (- 2 s) + e^{- s^{2}} (2 s + 0)$

Step 3.7

Add $2 s$ and $0$ .

$(s^{2} + 1) e^{- s^{2}} (- 2 s) + e^{- s^{2}} (2 s)$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$(s^{2} e^{- s^{2}} + 1 e^{- s^{2}}) (- 2 s) + e^{- s^{2}} (2 s)$

Step 4.2

Apply the distributive property.

$s^{2} e^{- s^{2}} (- 2 s) + 1 e^{- s^{2}} (- 2 s) + e^{- s^{2}} (2 s)$

Step 4.3

Combine terms.

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Step 4.3.1

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

$s^{1} s^{2} e^{- s^{2}} \cdot - 2 + 1 e^{- s^{2}} (- 2 s) + e^{- s^{2}} (2 s)$

Step 4.3.2

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

$s^{1 + 2} e^{- s^{2}} \cdot - 2 + 1 e^{- s^{2}} (- 2 s) + e^{- s^{2}} (2 s)$

Step 4.3.3

Add $1$ and $2$ .

$s^{3} e^{- s^{2}} \cdot - 2 + 1 e^{- s^{2}} (- 2 s) + e^{- s^{2}} (2 s)$

Step 4.3.4

Move $- 2$ to the left of $s^{3} e^{- s^{2}}$ .

$- 2 \cdot (s^{3} e^{- s^{2}}) + 1 e^{- s^{2}} (- 2 s) + e^{- s^{2}} (2 s)$

Step 4.3.5

Multiply $e^{- s^{2}}$ by $1$ .

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

Step 4.3.6

Move $e^{- s^{2}}$ .

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

Step 4.3.7

Add $- 2 s e^{- s^{2}}$ and $2 s e^{- s^{2}}$ .

$- 2 s^{3} e^{- s^{2}} + 0$

Step 4.3.8

Add $- 2 s^{3} e^{- s^{2}}$ and $0$ .

$- 2 s^{3} e^{- s^{2}}$

Step 4.4

Reorder the factors of $- 2 s^{3} e^{- s^{2}}$ .

$- 2 e^{- s^{2}} s^{3}$

Step 4.5

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

$- 2 s^{3} e^{- s^{2}}$