Calculus Examples

$f (x) = 55.36 e^{- 0.5 x} (- 26.9 x + 127.1)$

Step 1

Since $55.36$ is constant with respect to $x$ , the derivative of $55.36 e^{- 0.5 x} (- 26.9 x + 127.1)$ with respect to $x$ is $55.36 \frac{d}{d x} [e^{- 0.5 x} (- 26.9 x + 127.1)]$ .

$55.36 \frac{d}{d x} [e^{- 0.5 x} (- 26.9 x + 127.1)]$

Step 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 $f (x) = e^{- 0.5 x}$ and $g (x) = - 26.9 x + 127.1$ .

$55.36 (e^{- 0.5 x} \frac{d}{d x} [- 26.9 x + 127.1] + (- 26.9 x + 127.1) \frac{d}{d x} [e^{- 0.5 x}])$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $- 26.9 x + 127.1$ with respect to $x$ is $\frac{d}{d x} [- 26.9 x] + \frac{d}{d x} [127.1]$ .

$55.36 (e^{- 0.5 x} (\frac{d}{d x} [- 26.9 x] + \frac{d}{d x} [127.1]) + (- 26.9 x + 127.1) \frac{d}{d x} [e^{- 0.5 x}])$

Step 3.2

Since $- 26.9$ is constant with respect to $x$ , the derivative of $- 26.9 x$ with respect to $x$ is $- 26.9 \frac{d}{d x} [x]$ .

$55.36 (e^{- 0.5 x} (- 26.9 \frac{d}{d x} [x] + \frac{d}{d x} [127.1]) + (- 26.9 x + 127.1) \frac{d}{d x} [e^{- 0.5 x}])$

Step 3.3

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

$55.36 (e^{- 0.5 x} (- 26.9 \cdot 1 + \frac{d}{d x} [127.1]) + (- 26.9 x + 127.1) \frac{d}{d x} [e^{- 0.5 x}])$

Step 3.4

Multiply $- 26.9$ by $1$ .

$55.36 (e^{- 0.5 x} (- 26.9 + \frac{d}{d x} [127.1]) + (- 26.9 x + 127.1) \frac{d}{d x} [e^{- 0.5 x}])$

Step 3.5

Since $127.1$ is constant with respect to $x$ , the derivative of $127.1$ with respect to $x$ is $0$ .

$55.36 (e^{- 0.5 x} (- 26.9 + 0) + (- 26.9 x + 127.1) \frac{d}{d x} [e^{- 0.5 x}])$

Step 3.6

Simplify the expression.

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Step 3.6.1

Add $- 26.9$ and $0$ .

$55.36 (e^{- 0.5 x} \cdot - 26.9 + (- 26.9 x + 127.1) \frac{d}{d x} [e^{- 0.5 x}])$

Step 3.6.2

Move $- 26.9$ to the left of $e^{- 0.5 x}$ .

$55.36 (- 26.9 \cdot e^{- 0.5 x} + (- 26.9 x + 127.1) \frac{d}{d x} [e^{- 0.5 x}])$

Step 4

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 $g (x) = - 0.5 x$ .

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Step 4.1

To apply the Chain Rule, set $u$ as $- 0.5 x$ .

$55.36 (- 26.9 e^{- 0.5 x} + (- 26.9 x + 127.1) (\frac{d}{d u} [e^{u}] \frac{d}{d x} [- 0.5 x]))$

Step 4.2

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

$55.36 (- 26.9 e^{- 0.5 x} + (- 26.9 x + 127.1) (e^{u} \frac{d}{d x} [- 0.5 x]))$

Step 4.3

Replace all occurrences of $u$ with $- 0.5 x$ .

$55.36 (- 26.9 e^{- 0.5 x} + (- 26.9 x + 127.1) (e^{- 0.5 x} \frac{d}{d x} [- 0.5 x]))$

Step 5

Differentiate.

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Step 5.1

Since $- 0.5$ is constant with respect to $x$ , the derivative of $- 0.5 x$ with respect to $x$ is $- 0.5 \frac{d}{d x} [x]$ .

$55.36 (- 26.9 e^{- 0.5 x} + (- 26.9 x + 127.1) e^{- 0.5 x} (- 0.5 \frac{d}{d x} [x]))$

Step 5.2

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

$55.36 (- 26.9 e^{- 0.5 x} + (- 26.9 x + 127.1) e^{- 0.5 x} (- 0.5 \cdot 1))$

Step 5.3

Simplify the expression.

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Step 5.3.1

Multiply $- 0.5$ by $1$ .

$55.36 (- 26.9 e^{- 0.5 x} + (- 26.9 x + 127.1) e^{- 0.5 x} \cdot - 0.5)$

Step 5.3.2

Move $- 0.5$ to the left of $(- 26.9 x + 127.1) e^{- 0.5 x}$ .

$55.36 (- 26.9 e^{- 0.5 x} - 0.5 \cdot ((- 26.9 x + 127.1) e^{- 0.5 x}))$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$55.36 (- 26.9 e^{- 0.5 x} + (- 0.5 (- 26.9 x) - 0.5 \cdot 127.1) e^{- 0.5 x})$

Step 6.2

Apply the distributive property.

$55.36 (- 26.9 e^{- 0.5 x} - 0.5 (- 26.9 x) e^{- 0.5 x} - 0.5 \cdot 127.1 e^{- 0.5 x})$

Step 6.3

Apply the distributive property.

$55.36 (- 26.9 e^{- 0.5 x}) + 55.36 (- 0.5 (- 26.9 x) e^{- 0.5 x}) + 55.36 (- 0.5 \cdot 127.1 e^{- 0.5 x})$

Step 6.4

Combine terms.

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Step 6.4.1

Multiply $- 26.9$ by $55.36$ .

$- 1489.184 e^{- 0.5 x} + 55.36 (- 0.5 (- 26.9 x) e^{- 0.5 x}) + 55.36 (- 0.5 \cdot 127.1 e^{- 0.5 x})$

Step 6.4.2

Multiply $- 26.9$ by $- 0.5$ .

$- 1489.184 e^{- 0.5 x} + 55.36 (13.45 x e^{- 0.5 x}) + 55.36 (- 0.5 \cdot 127.1 e^{- 0.5 x})$

Step 6.4.3

Multiply $13.45$ by $55.36$ .

$- 1489.184 e^{- 0.5 x} + 744.592 (x e^{- 0.5 x}) + 55.36 (- 0.5 \cdot 127.1 e^{- 0.5 x})$

Step 6.4.4

Multiply $- 0.5$ by $127.1$ .

$- 1489.184 e^{- 0.5 x} + 744.592 x e^{- 0.5 x} + 55.36 (- 63.55 e^{- 0.5 x})$

Step 6.4.5

Multiply $- 63.55$ by $55.36$ .

$- 1489.184 e^{- 0.5 x} + 744.592 x e^{- 0.5 x} - 3518.128 e^{- 0.5 x}$

Step 6.4.6

Subtract $3518.128 e^{- 0.5 x}$ from $- 1489.184 e^{- 0.5 x}$ .

$744.592 x e^{- 0.5 x} - 5007.312 e^{- 0.5 x}$

Step 6.5

Reorder terms.

$744.592 e^{- 0.5 x} x - 5007.312 e^{- 0.5 x}$

Step 6.6

Reorder factors in $744.592 e^{- 0.5 x} x - 5007.312 e^{- 0.5 x}$ .

$744.592 x e^{- 0.5 x} - 5007.312 e^{- 0.5 x}$