Calculus Examples

$f (x) = (4 x^{2} - 8 x + 8) e^{x}$

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 $f (x) = 4 x^{2} - 8 x + 8$ and $g (x) = e^{x}$ .

$(4 x^{2} - 8 x + 8) \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [4 x^{2} - 8 x + 8]$

Step 2

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

$(4 x^{2} - 8 x + 8) e^{x} + e^{x} \frac{d}{d x} [4 x^{2} - 8 x + 8]$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $4 x^{2} - 8 x + 8$ with respect to $x$ is $\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [8]$ .

$(4 x^{2} - 8 x + 8) e^{x} + e^{x} (\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [8])$

Step 3.2

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{2}$ with respect to $x$ is $4 \frac{d}{d x} [x^{2}]$ .

$(4 x^{2} - 8 x + 8) e^{x} + e^{x} (4 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [8])$

Step 3.3

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

$(4 x^{2} - 8 x + 8) e^{x} + e^{x} (4 (2 x) + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [8])$

Step 3.4

Multiply $2$ by $4$ .

$(4 x^{2} - 8 x + 8) e^{x} + e^{x} (8 x + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [8])$

Step 3.5

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 x$ with respect to $x$ is $- 8 \frac{d}{d x} [x]$ .

$(4 x^{2} - 8 x + 8) e^{x} + e^{x} (8 x - 8 \frac{d}{d x} [x] + \frac{d}{d x} [8])$

Step 3.6

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

$(4 x^{2} - 8 x + 8) e^{x} + e^{x} (8 x - 8 \cdot 1 + \frac{d}{d x} [8])$

Step 3.7

Multiply $- 8$ by $1$ .

$(4 x^{2} - 8 x + 8) e^{x} + e^{x} (8 x - 8 + \frac{d}{d x} [8])$

Step 3.8

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$(4 x^{2} - 8 x + 8) e^{x} + e^{x} (8 x - 8 + 0)$

Step 3.9

Add $8 x - 8$ and $0$ .

$(4 x^{2} - 8 x + 8) e^{x} + e^{x} (8 x - 8)$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$4 x^{2} e^{x} - 8 x e^{x} + 8 e^{x} + e^{x} (8 x - 8)$

Step 4.2

Apply the distributive property.

$4 x^{2} e^{x} - 8 x e^{x} + 8 e^{x} + e^{x} (8 x) + e^{x} \cdot - 8$

Step 4.3

Combine terms.

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Step 4.3.1

Move $- 8$ to the left of $e^{x}$ .

$4 x^{2} e^{x} - 8 x e^{x} + 8 e^{x} + e^{x} (8 x) - 8 \cdot e^{x}$

Step 4.3.2

Add $- 8 x e^{x}$ and $e^{x} (8 x)$ .

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Step 4.3.2.1

Move $e^{x}$ .

$4 x^{2} e^{x} + 8 e^{x} - 8 x e^{x} + 8 x e^{x} - 8 e^{x}$

Step 4.3.2.2

Add $- 8 x e^{x}$ and $8 x e^{x}$ .

$4 x^{2} e^{x} + 8 e^{x} + 0 - 8 e^{x}$

Step 4.3.3

Add $4 x^{2} e^{x} + 8 e^{x}$ and $0$ .

$4 x^{2} e^{x} + 8 e^{x} - 8 e^{x}$

Step 4.3.4

Subtract $8 e^{x}$ from $8 e^{x}$ .

$4 x^{2} e^{x} + 0$

Step 4.3.5

Add $4 x^{2} e^{x}$ and $0$ .

$4 x^{2} e^{x}$

Step 4.4

Reorder the factors of $4 x^{2} e^{x}$ .

$4 e^{x} x^{2}$

Step 4.5

Reorder factors in $4 e^{x} x^{2}$ .

$4 x^{2} e^{x}$