Calculus Examples

$y = (13 x^{2} - 26 x + 26) e^{- 7 x}$

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 $f (x) = 13 x^{2} - 26 x + 26$ and $g (x) = e^{- 7 x}$ .

$(13 x^{2} - 26 x + 26) \frac{d}{d x} [e^{- 7 x}] + e^{- 7 x} \frac{d}{d x} [13 x^{2} - 26 x + 26]$

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 $g (x) = - 7 x$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $- 7 x$ .

$(13 x^{2} - 26 x + 26) (\frac{d}{d u} [e^{u}] \frac{d}{d x} [- 7 x]) + e^{- 7 x} \frac{d}{d x} [13 x^{2} - 26 x + 26]$

Step 2.2

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

$(13 x^{2} - 26 x + 26) (e^{u} \frac{d}{d x} [- 7 x]) + e^{- 7 x} \frac{d}{d x} [13 x^{2} - 26 x + 26]$

Step 2.3

Replace all occurrences of $u$ with $- 7 x$ .

$(13 x^{2} - 26 x + 26) (e^{- 7 x} \frac{d}{d x} [- 7 x]) + e^{- 7 x} \frac{d}{d x} [13 x^{2} - 26 x + 26]$

Step 3

Differentiate.

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Step 3.1

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7 x$ with respect to $x$ is $- 7 \frac{d}{d x} [x]$ .

$(13 x^{2} - 26 x + 26) e^{- 7 x} (- 7 \frac{d}{d x} [x]) + e^{- 7 x} \frac{d}{d x} [13 x^{2} - 26 x + 26]$

Step 3.2

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

$(13 x^{2} - 26 x + 26) e^{- 7 x} (- 7 \cdot 1) + e^{- 7 x} \frac{d}{d x} [13 x^{2} - 26 x + 26]$

Step 3.3

Simplify the expression.

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Step 3.3.1

Multiply $- 7$ by $1$ .

$(13 x^{2} - 26 x + 26) e^{- 7 x} \cdot - 7 + e^{- 7 x} \frac{d}{d x} [13 x^{2} - 26 x + 26]$

Step 3.3.2

Move $- 7$ to the left of $(13 x^{2} - 26 x + 26) e^{- 7 x}$ .

$- 7 \cdot ((13 x^{2} - 26 x + 26) e^{- 7 x}) + e^{- 7 x} \frac{d}{d x} [13 x^{2} - 26 x + 26]$

Step 3.4

By the Sum Rule, the derivative of $13 x^{2} - 26 x + 26$ with respect to $x$ is $\frac{d}{d x} [13 x^{2}] + \frac{d}{d x} [- 26 x] + \frac{d}{d x} [26]$ .

$- 7 (13 x^{2} - 26 x + 26) e^{- 7 x} + e^{- 7 x} (\frac{d}{d x} [13 x^{2}] + \frac{d}{d x} [- 26 x] + \frac{d}{d x} [26])$

Step 3.5

Since $13$ is constant with respect to $x$ , the derivative of $13 x^{2}$ with respect to $x$ is $13 \frac{d}{d x} [x^{2}]$ .

$- 7 (13 x^{2} - 26 x + 26) e^{- 7 x} + e^{- 7 x} (13 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 26 x] + \frac{d}{d x} [26])$

Step 3.6

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

$- 7 (13 x^{2} - 26 x + 26) e^{- 7 x} + e^{- 7 x} (13 (2 x) + \frac{d}{d x} [- 26 x] + \frac{d}{d x} [26])$

Step 3.7

Multiply $2$ by $13$ .

$- 7 (13 x^{2} - 26 x + 26) e^{- 7 x} + e^{- 7 x} (26 x + \frac{d}{d x} [- 26 x] + \frac{d}{d x} [26])$

Step 3.8

Since $- 26$ is constant with respect to $x$ , the derivative of $- 26 x$ with respect to $x$ is $- 26 \frac{d}{d x} [x]$ .

$- 7 (13 x^{2} - 26 x + 26) e^{- 7 x} + e^{- 7 x} (26 x - 26 \frac{d}{d x} [x] + \frac{d}{d x} [26])$

Step 3.9

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

$- 7 (13 x^{2} - 26 x + 26) e^{- 7 x} + e^{- 7 x} (26 x - 26 \cdot 1 + \frac{d}{d x} [26])$

Step 3.10

Multiply $- 26$ by $1$ .

$- 7 (13 x^{2} - 26 x + 26) e^{- 7 x} + e^{- 7 x} (26 x - 26 + \frac{d}{d x} [26])$

Step 3.11

Since $26$ is constant with respect to $x$ , the derivative of $26$ with respect to $x$ is $0$ .

$- 7 (13 x^{2} - 26 x + 26) e^{- 7 x} + e^{- 7 x} (26 x - 26 + 0)$

Step 3.12

Add $26 x - 26$ and $0$ .

$- 7 (13 x^{2} - 26 x + 26) e^{- 7 x} + e^{- 7 x} (26 x - 26)$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$(- 7 (13 x^{2}) - 7 (- 26 x) - 7 \cdot 26) e^{- 7 x} + e^{- 7 x} (26 x - 26)$

Step 4.2

Apply the distributive property.

$- 7 (13 x^{2}) e^{- 7 x} - 7 (- 26 x) e^{- 7 x} - 7 \cdot 26 e^{- 7 x} + e^{- 7 x} (26 x - 26)$

Step 4.3

Apply the distributive property.

$- 7 (13 x^{2}) e^{- 7 x} - 7 (- 26 x) e^{- 7 x} - 7 \cdot 26 e^{- 7 x} + e^{- 7 x} (26 x) + e^{- 7 x} \cdot - 26$

Step 4.4

Combine terms.

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Step 4.4.1

Multiply $13$ by $- 7$ .

$- 91 x^{2} e^{- 7 x} - 7 (- 26 x) e^{- 7 x} - 7 \cdot 26 e^{- 7 x} + e^{- 7 x} (26 x) + e^{- 7 x} \cdot - 26$

Step 4.4.2

Multiply $- 26$ by $- 7$ .

$- 91 x^{2} e^{- 7 x} + 182 x e^{- 7 x} - 7 \cdot 26 e^{- 7 x} + e^{- 7 x} (26 x) + e^{- 7 x} \cdot - 26$

Step 4.4.3

Multiply $- 7$ by $26$ .

$- 91 x^{2} e^{- 7 x} + 182 x e^{- 7 x} - 182 e^{- 7 x} + e^{- 7 x} (26 x) + e^{- 7 x} \cdot - 26$

Step 4.4.4

Move $- 26$ to the left of $e^{- 7 x}$ .

$- 91 x^{2} e^{- 7 x} + 182 x e^{- 7 x} - 182 e^{- 7 x} + e^{- 7 x} (26 x) - 26 \cdot e^{- 7 x}$

Step 4.4.5

Add $182 x e^{- 7 x}$ and $e^{- 7 x} (26 x)$ .

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Step 4.4.5.1

Move $e^{- 7 x}$ .

$- 91 x^{2} e^{- 7 x} + 182 x e^{- 7 x} + 26 x e^{- 7 x} - 182 e^{- 7 x} - 26 e^{- 7 x}$

Step 4.4.5.2

Add $182 x e^{- 7 x}$ and $26 x e^{- 7 x}$ .

$- 91 x^{2} e^{- 7 x} + 208 x e^{- 7 x} - 182 e^{- 7 x} - 26 e^{- 7 x}$

Step 4.4.6

Subtract $26 e^{- 7 x}$ from $- 182 e^{- 7 x}$ .

$- 91 x^{2} e^{- 7 x} + 208 x e^{- 7 x} - 208 e^{- 7 x}$

Step 4.5

Reorder terms.

$- 91 e^{- 7 x} x^{2} + 208 e^{- 7 x} x - 208 e^{- 7 x}$

Step 4.6

Reorder factors in $- 91 e^{- 7 x} x^{2} + 208 e^{- 7 x} x - 208 e^{- 7 x}$ .

$- 91 x^{2} e^{- 7 x} + 208 x e^{- 7 x} - 208 e^{- 7 x}$