Find the Derivative - d/dx y=(11x^2-22x+22)e^(-3x)

$y = (11 x^{2} - 22 x + 22) e^{- 3 x}$

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 $f (x) = 11 x^{2} - 22 x + 22$ and $g (x) = e^{- 3 x}$ .

$(11 x^{2} - 22 x + 22) \frac{d}{d x} [e^{- 3 x}] + e^{- 3 x} \frac{d}{d x} [11 x^{2} - 22 x + 22]$

Step 2

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 $g (x) = - 3 x$ .

To apply the Chain Rule, set $u$ as $- 3 x$ .

$(11 x^{2} - 22 x + 22) (\frac{d}{d u} [e^{u}] \frac{d}{d x} [- 3 x]) + e^{- 3 x} \frac{d}{d x} [11 x^{2} - 22 x + 22]$

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

$(11 x^{2} - 22 x + 22) (e^{u} \frac{d}{d x} [- 3 x]) + e^{- 3 x} \frac{d}{d x} [11 x^{2} - 22 x + 22]$

Replace all occurrences of $u$ with $- 3 x$ .

$(11 x^{2} - 22 x + 22) (e^{- 3 x} \frac{d}{d x} [- 3 x]) + e^{- 3 x} \frac{d}{d x} [11 x^{2} - 22 x + 22]$

Step 3

Differentiate.

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x$ with respect to $x$ is $- 3 \frac{d}{d x} [x]$ .

$(11 x^{2} - 22 x + 22) e^{- 3 x} (- 3 \frac{d}{d x} [x]) + e^{- 3 x} \frac{d}{d x} [11 x^{2} - 22 x + 22]$

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

$(11 x^{2} - 22 x + 22) e^{- 3 x} (- 3 \cdot 1) + e^{- 3 x} \frac{d}{d x} [11 x^{2} - 22 x + 22]$

Simplify the expression.

Multiply $- 3$ by $1$ .

$(11 x^{2} - 22 x + 22) e^{- 3 x} \cdot - 3 + e^{- 3 x} \frac{d}{d x} [11 x^{2} - 22 x + 22]$

Move $- 3$ to the left of $(11 x^{2} - 22 x + 22) e^{- 3 x}$ .

$- 3 \cdot ((11 x^{2} - 22 x + 22) e^{- 3 x}) + e^{- 3 x} \frac{d}{d x} [11 x^{2} - 22 x + 22]$

By the Sum Rule, the derivative of $11 x^{2} - 22 x + 22$ with respect to $x$ is $\frac{d}{d x} [11 x^{2}] + \frac{d}{d x} [- 22 x] + \frac{d}{d x} [22]$ .

$- 3 (11 x^{2} - 22 x + 22) e^{- 3 x} + e^{- 3 x} (\frac{d}{d x} [11 x^{2}] + \frac{d}{d x} [- 22 x] + \frac{d}{d x} [22])$

Since $11$ is constant with respect to $x$ , the derivative of $11 x^{2}$ with respect to $x$ is $11 \frac{d}{d x} [x^{2}]$ .

$- 3 (11 x^{2} - 22 x + 22) e^{- 3 x} + e^{- 3 x} (11 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 22 x] + \frac{d}{d x} [22])$

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

$- 3 (11 x^{2} - 22 x + 22) e^{- 3 x} + e^{- 3 x} (11 (2 x) + \frac{d}{d x} [- 22 x] + \frac{d}{d x} [22])$

Multiply $2$ by $11$ .

$- 3 (11 x^{2} - 22 x + 22) e^{- 3 x} + e^{- 3 x} (22 x + \frac{d}{d x} [- 22 x] + \frac{d}{d x} [22])$

Since $- 22$ is constant with respect to $x$ , the derivative of $- 22 x$ with respect to $x$ is $- 22 \frac{d}{d x} [x]$ .

$- 3 (11 x^{2} - 22 x + 22) e^{- 3 x} + e^{- 3 x} (22 x - 22 \frac{d}{d x} [x] + \frac{d}{d x} [22])$

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

$- 3 (11 x^{2} - 22 x + 22) e^{- 3 x} + e^{- 3 x} (22 x - 22 \cdot 1 + \frac{d}{d x} [22])$

Multiply $- 22$ by $1$ .

$- 3 (11 x^{2} - 22 x + 22) e^{- 3 x} + e^{- 3 x} (22 x - 22 + \frac{d}{d x} [22])$

Since $22$ is constant with respect to $x$ , the derivative of $22$ with respect to $x$ is $0$ .

$- 3 (11 x^{2} - 22 x + 22) e^{- 3 x} + e^{- 3 x} (22 x - 22 + 0)$

Add $22 x - 22$ and $0$ .

$- 3 (11 x^{2} - 22 x + 22) e^{- 3 x} + e^{- 3 x} (22 x - 22)$

Step 4

Simplify.

Apply the distributive property.

$(- 3 (11 x^{2}) - 3 (- 22 x) - 3 \cdot 22) e^{- 3 x} + e^{- 3 x} (22 x - 22)$

Apply the distributive property.

$- 3 (11 x^{2}) e^{- 3 x} - 3 (- 22 x) e^{- 3 x} - 3 \cdot 22 e^{- 3 x} + e^{- 3 x} (22 x - 22)$

Apply the distributive property.

$- 3 (11 x^{2}) e^{- 3 x} - 3 (- 22 x) e^{- 3 x} - 3 \cdot 22 e^{- 3 x} + e^{- 3 x} (22 x) + e^{- 3 x} \cdot - 22$

Combine terms.

Multiply $11$ by $- 3$ .

$- 33 x^{2} e^{- 3 x} - 3 (- 22 x) e^{- 3 x} - 3 \cdot 22 e^{- 3 x} + e^{- 3 x} (22 x) + e^{- 3 x} \cdot - 22$

Multiply $- 22$ by $- 3$ .

$- 33 x^{2} e^{- 3 x} + 66 x e^{- 3 x} - 3 \cdot 22 e^{- 3 x} + e^{- 3 x} (22 x) + e^{- 3 x} \cdot - 22$

Multiply $- 3$ by $22$ .

$- 33 x^{2} e^{- 3 x} + 66 x e^{- 3 x} - 66 e^{- 3 x} + e^{- 3 x} (22 x) + e^{- 3 x} \cdot - 22$

Move $- 22$ to the left of $e^{- 3 x}$ .

$- 33 x^{2} e^{- 3 x} + 66 x e^{- 3 x} - 66 e^{- 3 x} + e^{- 3 x} (22 x) - 22 \cdot e^{- 3 x}$

Add $66 x e^{- 3 x}$ and $e^{- 3 x} (22 x)$ .

Move $e^{- 3 x}$ .

$- 33 x^{2} e^{- 3 x} + 66 x e^{- 3 x} + 22 x e^{- 3 x} - 66 e^{- 3 x} - 22 e^{- 3 x}$

Add $66 x e^{- 3 x}$ and $22 x e^{- 3 x}$ .

$- 33 x^{2} e^{- 3 x} + 88 x e^{- 3 x} - 66 e^{- 3 x} - 22 e^{- 3 x}$

Subtract $22 e^{- 3 x}$ from $- 66 e^{- 3 x}$ .

$- 33 x^{2} e^{- 3 x} + 88 x e^{- 3 x} - 88 e^{- 3 x}$

Reorder terms.

$- 33 e^{- 3 x} x^{2} + 88 e^{- 3 x} x - 88 e^{- 3 x}$

Reorder factors in $- 33 e^{- 3 x} x^{2} + 88 e^{- 3 x} x - 88 e^{- 3 x}$ .

$- 33 x^{2} e^{- 3 x} + 88 x e^{- 3 x} - 88 e^{- 3 x}$

Find the Derivative - d/dx y=(11x^2-22x+22)e^(-3x)

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