Calculus Examples

$f (x) = (10 - 5 x) e^{x}$

Step 1

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 $f (x) = 10 - 5 x$ and $g (x) = e^{x}$ .

$(10 - 5 x) \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [10 - 5 x]$

Step 2

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

$(10 - 5 x) e^{x} + e^{x} \frac{d}{d x} [10 - 5 x]$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $10 - 5 x$ with respect to $x$ is $\frac{d}{d x} [10] + \frac{d}{d x} [- 5 x]$ .

$(10 - 5 x) e^{x} + e^{x} (\frac{d}{d x} [10] + \frac{d}{d x} [- 5 x])$

Step 3.2

Since $10$ is constant with respect to $x$ , the derivative of $10$ with respect to $x$ is $0$ .

$(10 - 5 x) e^{x} + e^{x} (0 + \frac{d}{d x} [- 5 x])$

Step 3.3

Add $0$ and $\frac{d}{d x} [- 5 x]$ .

$(10 - 5 x) e^{x} + e^{x} \frac{d}{d x} [- 5 x]$

Step 3.4

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x$ with respect to $x$ is $- 5 \frac{d}{d x} [x]$ .

$(10 - 5 x) e^{x} + e^{x} (- 5 \frac{d}{d x} [x])$

Step 3.5

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

$(10 - 5 x) e^{x} + e^{x} (- 5 \cdot 1)$

Step 3.6

Simplify the expression.

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Step 3.6.1

Multiply $- 5$ by $1$ .

$(10 - 5 x) e^{x} + e^{x} \cdot - 5$

Step 3.6.2

Move $- 5$ to the left of $e^{x}$ .

$(10 - 5 x) e^{x} - 5 \cdot e^{x}$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$10 e^{x} - 5 x e^{x} - 5 e^{x}$

Step 4.2

Subtract $5 e^{x}$ from $10 e^{x}$ .

$- 5 x e^{x} + 5 e^{x}$

Step 4.3

Reorder terms.

$- 5 e^{x} x + 5 e^{x}$

Step 4.4

Reorder factors in $- 5 e^{x} x + 5 e^{x}$ .

$- 5 x e^{x} + 5 e^{x}$