Calculus Examples

$y = (1 + x - x^{2}) e^{x}$

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 $f (x) = 1 + x - x^{2}$ and $g (x) = e^{x}$ .

$(1 + x - x^{2}) \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [1 + x - x^{2}]$

Step 2

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

$(1 + x - x^{2}) e^{x} + e^{x} \frac{d}{d x} [1 + x - x^{2}]$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $1 + x - x^{2}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [x] + \frac{d}{d x} [- x^{2}]$ .

$(1 + x - x^{2}) e^{x} + e^{x} (\frac{d}{d x} [1] + \frac{d}{d x} [x] + \frac{d}{d x} [- x^{2}])$

Step 3.2

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

$(1 + x - x^{2}) e^{x} + e^{x} (0 + \frac{d}{d x} [x] + \frac{d}{d x} [- x^{2}])$

Step 3.3

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

$(1 + x - x^{2}) e^{x} + e^{x} (\frac{d}{d x} [x] + \frac{d}{d x} [- x^{2}])$

Step 3.4

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

$(1 + x - x^{2}) e^{x} + e^{x} (1 + \frac{d}{d x} [- x^{2}])$

Step 3.5

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$(1 + x - x^{2}) e^{x} + e^{x} (1 - \frac{d}{d x} [x^{2}])$

Step 3.6

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

$(1 + x - x^{2}) e^{x} + e^{x} (1 - (2 x))$

Step 3.7

Multiply $2$ by $- 1$ .

$(1 + x - x^{2}) e^{x} + e^{x} (1 - 2 x)$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$1 e^{x} + x e^{x} - x^{2} e^{x} + e^{x} (1 - 2 x)$

Step 4.2

Apply the distributive property.

$1 e^{x} + x e^{x} - x^{2} e^{x} + e^{x} \cdot 1 + e^{x} (- 2 x)$

Step 4.3

Combine terms.

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Step 4.3.1

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

$e^{x} + x e^{x} - x^{2} e^{x} + e^{x} \cdot 1 + e^{x} (- 2 x)$

Step 4.3.2

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

$e^{x} + x e^{x} - x^{2} e^{x} + e^{x} + e^{x} (- 2 x)$

Step 4.3.3

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

$x e^{x} - x^{2} e^{x} + 2 e^{x} + e^{x} (- 2 x)$

Step 4.3.4

Add $x e^{x}$ and $e^{x} (- 2 x)$ .

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Step 4.3.4.1

Move $e^{x}$ .

$x e^{x} - 2 x e^{x} - x^{2} e^{x} + 2 e^{x}$

Step 4.3.4.2

Subtract $2 x e^{x}$ from $x e^{x}$ .

$- x e^{x} - x^{2} e^{x} + 2 e^{x}$

Step 4.4

Reorder terms.

$- e^{x} x - e^{x} x^{2} + 2 e^{x}$

Step 4.5

Reorder factors in $- e^{x} x - e^{x} x^{2} + 2 e^{x}$ .

$- x e^{x} - x^{2} e^{x} + 2 e^{x}$