Calculus Examples

$(17 w^{2} + 11) e^{w^{2}}$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d w} [f (w) g (w)]$ is $f (w) \frac{d}{d w} [g (w)] + g (w) \frac{d}{d w} [f (w)]$ where $f (w) = 17 w^{2} + 11$ and $g (w) = e^{w^{2}}$ .

$(17 w^{2} + 11) \frac{d}{d w} [e^{w^{2}}] + e^{w^{2}} \frac{d}{d w} [17 w^{2} + 11]$

Step 2

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

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Step 2.1

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

$(17 w^{2} + 11) (\frac{d}{d u} [e^{u}] \frac{d}{d w} [w^{2}]) + e^{w^{2}} \frac{d}{d w} [17 w^{2} + 11]$

Step 2.2

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

$(17 w^{2} + 11) (e^{u} \frac{d}{d w} [w^{2}]) + e^{w^{2}} \frac{d}{d w} [17 w^{2} + 11]$

Step 2.3

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

$(17 w^{2} + 11) (e^{w^{2}} \frac{d}{d w} [w^{2}]) + e^{w^{2}} \frac{d}{d w} [17 w^{2} + 11]$

Step 3

Differentiate.

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Step 3.1

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

$(17 w^{2} + 11) e^{w^{2}} (2 w) + e^{w^{2}} \frac{d}{d w} [17 w^{2} + 11]$

Step 3.2

By the Sum Rule, the derivative of $17 w^{2} + 11$ with respect to $w$ is $\frac{d}{d w} [17 w^{2}] + \frac{d}{d w} [11]$ .

$(17 w^{2} + 11) e^{w^{2}} (2 w) + e^{w^{2}} (\frac{d}{d w} [17 w^{2}] + \frac{d}{d w} [11])$

Step 3.3

Since $17$ is constant with respect to $w$ , the derivative of $17 w^{2}$ with respect to $w$ is $17 \frac{d}{d w} [w^{2}]$ .

$(17 w^{2} + 11) e^{w^{2}} (2 w) + e^{w^{2}} (17 \frac{d}{d w} [w^{2}] + \frac{d}{d w} [11])$

Step 3.4

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

$(17 w^{2} + 11) e^{w^{2}} (2 w) + e^{w^{2}} (17 (2 w) + \frac{d}{d w} [11])$

Step 3.5

Multiply $2$ by $17$ .

$(17 w^{2} + 11) e^{w^{2}} (2 w) + e^{w^{2}} (34 w + \frac{d}{d w} [11])$

Step 3.6

Since $11$ is constant with respect to $w$ , the derivative of $11$ with respect to $w$ is $0$ .

$(17 w^{2} + 11) e^{w^{2}} (2 w) + e^{w^{2}} (34 w + 0)$

Step 3.7

Add $34 w$ and $0$ .

$(17 w^{2} + 11) e^{w^{2}} (2 w) + e^{w^{2}} (34 w)$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$(17 w^{2} e^{w^{2}} + 11 e^{w^{2}}) (2 w) + e^{w^{2}} (34 w)$

Step 4.2

Apply the distributive property.

$17 w^{2} e^{w^{2}} (2 w) + 11 e^{w^{2}} (2 w) + e^{w^{2}} (34 w)$

Step 4.3

Combine terms.

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Step 4.3.1

Multiply $2$ by $17$ .

$34 w^{2} e^{w^{2}} w + 11 e^{w^{2}} (2 w) + e^{w^{2}} (34 w)$

Step 4.3.2

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

$34 (w^{1} w^{2}) e^{w^{2}} + 11 e^{w^{2}} (2 w) + e^{w^{2}} (34 w)$

Step 4.3.3

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

$34 w^{1 + 2} e^{w^{2}} + 11 e^{w^{2}} (2 w) + e^{w^{2}} (34 w)$

Step 4.3.4

Add $1$ and $2$ .

$34 w^{3} e^{w^{2}} + 11 e^{w^{2}} (2 w) + e^{w^{2}} (34 w)$

Step 4.3.5

Multiply $2$ by $11$ .

$34 w^{3} e^{w^{2}} + 22 e^{w^{2}} w + e^{w^{2}} (34 w)$

Step 4.3.6

Move $e^{w^{2}}$ .

$34 w^{3} e^{w^{2}} + 22 w e^{w^{2}} + 34 w e^{w^{2}}$

Step 4.3.7

Add $22 w e^{w^{2}}$ and $34 w e^{w^{2}}$ .

$34 w^{3} e^{w^{2}} + 56 w e^{w^{2}}$

Step 4.4

Reorder terms.

$34 e^{w^{2}} w^{3} + 56 e^{w^{2}} w$

Step 4.5

Reorder factors in $34 e^{w^{2}} w^{3} + 56 e^{w^{2}} w$ .

$34 w^{3} e^{w^{2}} + 56 w e^{w^{2}}$