Calculus Examples

$(s^{8} + 1) e^{- s^{8}}$

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 $f (s) = s^{8} + 1$ and $g (s) = e^{- s^{8}}$ .

$(s^{8} + 1) \frac{d}{d s} [e^{- s^{8}}] + e^{- s^{8}} \frac{d}{d s} [s^{8} + 1]$

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 $g (s) = - s^{8}$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $- s^{8}$ .

$(s^{8} + 1) (\frac{d}{d u} [e^{u}] \frac{d}{d s} [- s^{8}]) + e^{- s^{8}} \frac{d}{d s} [s^{8} + 1]$

Step 2.2

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

$(s^{8} + 1) (e^{u} \frac{d}{d s} [- s^{8}]) + e^{- s^{8}} \frac{d}{d s} [s^{8} + 1]$

Step 2.3

Replace all occurrences of $u$ with $- s^{8}$ .

$(s^{8} + 1) (e^{- s^{8}} \frac{d}{d s} [- s^{8}]) + e^{- s^{8}} \frac{d}{d s} [s^{8} + 1]$

Step 3

Differentiate.

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Step 3.1

Since $- 1$ is constant with respect to $s$ , the derivative of $- s^{8}$ with respect to $s$ is $- \frac{d}{d s} [s^{8}]$ .

$(s^{8} + 1) e^{- s^{8}} (- \frac{d}{d s} [s^{8}]) + e^{- s^{8}} \frac{d}{d s} [s^{8} + 1]$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d s} [s^{n}]$ is $n s^{n - 1}$ where $n = 8$ .

$(s^{8} + 1) e^{- s^{8}} (- (8 s^{7})) + e^{- s^{8}} \frac{d}{d s} [s^{8} + 1]$

Step 3.3

Multiply $8$ by $- 1$ .

$(s^{8} + 1) e^{- s^{8}} (- 8 s^{7}) + e^{- s^{8}} \frac{d}{d s} [s^{8} + 1]$

Step 3.4

By the Sum Rule, the derivative of $s^{8} + 1$ with respect to $s$ is $\frac{d}{d s} [s^{8}] + \frac{d}{d s} [1]$ .

$(s^{8} + 1) e^{- s^{8}} (- 8 s^{7}) + e^{- s^{8}} (\frac{d}{d s} [s^{8}] + \frac{d}{d s} [1])$

Step 3.5

Differentiate using the Power Rule which states that $\frac{d}{d s} [s^{n}]$ is $n s^{n - 1}$ where $n = 8$ .

$(s^{8} + 1) e^{- s^{8}} (- 8 s^{7}) + e^{- s^{8}} (8 s^{7} + \frac{d}{d s} [1])$

Step 3.6

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

$(s^{8} + 1) e^{- s^{8}} (- 8 s^{7}) + e^{- s^{8}} (8 s^{7} + 0)$

Step 3.7

Add $8 s^{7}$ and $0$ .

$(s^{8} + 1) e^{- s^{8}} (- 8 s^{7}) + e^{- s^{8}} (8 s^{7})$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$(s^{8} e^{- s^{8}} + 1 e^{- s^{8}}) (- 8 s^{7}) + e^{- s^{8}} (8 s^{7})$

Step 4.2

Apply the distributive property.

$s^{8} e^{- s^{8}} (- 8 s^{7}) + 1 e^{- s^{8}} (- 8 s^{7}) + e^{- s^{8}} (8 s^{7})$

Step 4.3

Combine terms.

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Step 4.3.1

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

$s^{7 + 8} e^{- s^{8}} \cdot - 8 + 1 e^{- s^{8}} (- 8 s^{7}) + e^{- s^{8}} (8 s^{7})$

Step 4.3.2

Add $7$ and $8$ .

$s^{15} e^{- s^{8}} \cdot - 8 + 1 e^{- s^{8}} (- 8 s^{7}) + e^{- s^{8}} (8 s^{7})$

Step 4.3.3

Move $- 8$ to the left of $s^{15} e^{- s^{8}}$ .

$- 8 \cdot (s^{15} e^{- s^{8}}) + 1 e^{- s^{8}} (- 8 s^{7}) + e^{- s^{8}} (8 s^{7})$

Step 4.3.4

Multiply $e^{- s^{8}}$ by $1$ .

$- 8 s^{15} e^{- s^{8}} + e^{- s^{8}} (- 8 s^{7}) + e^{- s^{8}} (8 s^{7})$

Step 4.3.5

Move $e^{- s^{8}}$ .

$- 8 s^{15} e^{- s^{8}} - 8 s^{7} e^{- s^{8}} + 8 s^{7} e^{- s^{8}}$

Step 4.3.6

Add $- 8 s^{7} e^{- s^{8}}$ and $8 s^{7} e^{- s^{8}}$ .

$- 8 s^{15} e^{- s^{8}} + 0$

Step 4.3.7

Add $- 8 s^{15} e^{- s^{8}}$ and $0$ .

$- 8 s^{15} e^{- s^{8}}$

Step 4.4

Reorder the factors of $- 8 s^{15} e^{- s^{8}}$ .

$- 8 e^{- s^{8}} s^{15}$

Step 4.5

Reorder factors in $- 8 e^{- s^{8}} s^{15}$ .

$- 8 s^{15} e^{- s^{8}}$