Calculus Examples

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

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 $f (x) = x - 1$ and $g (x) = e^{x}$ .

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

Step 2

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

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

$(x - 1) e^{x} + e^{x} (1 + 0)$

Step 3.4

Simplify the expression.

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Step 3.4.1

Add $1$ and $0$ .

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

Step 3.4.2

Multiply $e^{x}$ by $1$ .

$(x - 1) e^{x} + e^{x}$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$x e^{x} - 1 e^{x} + e^{x}$

Step 4.2

Combine terms.

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Step 4.2.1

Rewrite $- 1 e^{x}$ as $- e^{x}$ .

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

Step 4.2.2

Add $- e^{x}$ and $e^{x}$ .

$x e^{x} + 0$

Step 4.2.3

Add $x e^{x}$ and $0$ .

$x e^{x}$

Step 4.3

Reorder the factors of $x e^{x}$ .

$e^{x} x$

Step 4.4

Reorder factors in $e^{x} x$ .

$x e^{x}$