Calculus Examples

$(x^{2} - 8 x + 8) e^{\frac{9 x}{8}}$

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 $f (x) = x^{2} - 8 x + 8$ and $g (x) = e^{\frac{9 x}{8}}$ .

$(x^{2} - 8 x + 8) \frac{d}{d x} [e^{\frac{9 x}{8}}] + e^{\frac{9 x}{8}} \frac{d}{d x} [x^{2} - 8 x + 8]$

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 $g (x) = \frac{9 x}{8}$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $\frac{9 x}{8}$ .

$(x^{2} - 8 x + 8) (\frac{d}{d u} [e^{u}] \frac{d}{d x} [\frac{9 x}{8}]) + e^{\frac{9 x}{8}} \frac{d}{d x} [x^{2} - 8 x + 8]$

Step 2.2

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

$(x^{2} - 8 x + 8) (e^{u} \frac{d}{d x} [\frac{9 x}{8}]) + e^{\frac{9 x}{8}} \frac{d}{d x} [x^{2} - 8 x + 8]$

Step 2.3

Replace all occurrences of $u$ with $\frac{9 x}{8}$ .

$(x^{2} - 8 x + 8) (e^{\frac{9 x}{8}} \frac{d}{d x} [\frac{9 x}{8}]) + e^{\frac{9 x}{8}} \frac{d}{d x} [x^{2} - 8 x + 8]$

Step 3

Differentiate.

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Step 3.1

Since $\frac{9}{8}$ is constant with respect to $x$ , the derivative of $\frac{9 x}{8}$ with respect to $x$ is $\frac{9}{8} \frac{d}{d x} [x]$ .

$(x^{2} - 8 x + 8) e^{\frac{9 x}{8}} (\frac{9}{8} \frac{d}{d x} [x]) + e^{\frac{9 x}{8}} \frac{d}{d x} [x^{2} - 8 x + 8]$

Step 3.2

Combine $\frac{9}{8}$ and $e^{\frac{9 x}{8}}$ .

$(x^{2} - 8 x + 8) \frac{9 e^{\frac{9 x}{8}}}{8} \frac{d}{d x} [x] + e^{\frac{9 x}{8}} \frac{d}{d x} [x^{2} - 8 x + 8]$

Step 3.3

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

$(x^{2} - 8 x + 8) \frac{9 e^{\frac{9 x}{8}}}{8} \cdot 1 + e^{\frac{9 x}{8}} \frac{d}{d x} [x^{2} - 8 x + 8]$

Step 3.4

Multiply $x^{2} - 8 x + 8$ by $1$ .

$(x^{2} - 8 x + 8) \frac{9 e^{\frac{9 x}{8}}}{8} + e^{\frac{9 x}{8}} \frac{d}{d x} [x^{2} - 8 x + 8]$

Step 3.5

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

$(x^{2} - 8 x + 8) \frac{9 e^{\frac{9 x}{8}}}{8} + e^{\frac{9 x}{8}} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [8])$

Step 3.6

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

$(x^{2} - 8 x + 8) \frac{9 e^{\frac{9 x}{8}}}{8} + e^{\frac{9 x}{8}} (2 x + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [8])$

Step 3.7

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

$(x^{2} - 8 x + 8) \frac{9 e^{\frac{9 x}{8}}}{8} + e^{\frac{9 x}{8}} (2 x - 8 \frac{d}{d x} [x] + \frac{d}{d x} [8])$

Step 3.8

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

$(x^{2} - 8 x + 8) \frac{9 e^{\frac{9 x}{8}}}{8} + e^{\frac{9 x}{8}} (2 x - 8 \cdot 1 + \frac{d}{d x} [8])$

Step 3.9

Multiply $- 8$ by $1$ .

$(x^{2} - 8 x + 8) \frac{9 e^{\frac{9 x}{8}}}{8} + e^{\frac{9 x}{8}} (2 x - 8 + \frac{d}{d x} [8])$

Step 3.10

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

$(x^{2} - 8 x + 8) \frac{9 e^{\frac{9 x}{8}}}{8} + e^{\frac{9 x}{8}} (2 x - 8 + 0)$

Step 3.11

Add $2 x - 8$ and $0$ .

$(x^{2} - 8 x + 8) \frac{9 e^{\frac{9 x}{8}}}{8} + e^{\frac{9 x}{8}} (2 x - 8)$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$x^{2} \frac{9 e^{\frac{9 x}{8}}}{8} - 8 x \frac{9 e^{\frac{9 x}{8}}}{8} + 8 \frac{9 e^{\frac{9 x}{8}}}{8} + e^{\frac{9 x}{8}} (2 x - 8)$

Step 4.2

Apply the distributive property.

$x^{2} \frac{9 e^{\frac{9 x}{8}}}{8} - 8 x \frac{9 e^{\frac{9 x}{8}}}{8} + 8 \frac{9 e^{\frac{9 x}{8}}}{8} + e^{\frac{9 x}{8}} (2 x) + e^{\frac{9 x}{8}} \cdot - 8$

Step 4.3

Combine terms.

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Step 4.3.1

Combine $x^{2}$ and $\frac{9 e^{\frac{9 x}{8}}}{8}$ .

$\frac{x^{2} (9 e^{\frac{9 x}{8}})}{8} - 8 x \frac{9 e^{\frac{9 x}{8}}}{8} + 8 \frac{9 e^{\frac{9 x}{8}}}{8} + e^{\frac{9 x}{8}} (2 x) + e^{\frac{9 x}{8}} \cdot - 8$

Step 4.3.2

Combine $- 8$ and $\frac{9 e^{\frac{9 x}{8}}}{8}$ .

$\frac{x^{2} (9 e^{\frac{9 x}{8}})}{8} + x \frac{- 8 (9 e^{\frac{9 x}{8}})}{8} + 8 \frac{9 e^{\frac{9 x}{8}}}{8} + e^{\frac{9 x}{8}} (2 x) + e^{\frac{9 x}{8}} \cdot - 8$

Step 4.3.3

Multiply $9$ by $- 8$ .

$\frac{x^{2} (9 e^{\frac{9 x}{8}})}{8} + x \frac{- 72 e^{\frac{9 x}{8}}}{8} + 8 \frac{9 e^{\frac{9 x}{8}}}{8} + e^{\frac{9 x}{8}} (2 x) + e^{\frac{9 x}{8}} \cdot - 8$

Step 4.3.4

Combine $x$ and $\frac{- 72 e^{\frac{9 x}{8}}}{8}$ .

$\frac{x^{2} (9 e^{\frac{9 x}{8}})}{8} + \frac{x (- 72 e^{\frac{9 x}{8}})}{8} + 8 \frac{9 e^{\frac{9 x}{8}}}{8} + e^{\frac{9 x}{8}} (2 x) + e^{\frac{9 x}{8}} \cdot - 8$

Step 4.3.5

Cancel the common factor of $- 72$ and $8$ .

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Step 4.3.5.1

Factor $8$ out of $x (- 72 e^{\frac{9 x}{8}})$ .

$\frac{x^{2} (9 e^{\frac{9 x}{8}})}{8} + \frac{8 (x (- 9 e^{\frac{9 x}{8}}))}{8} + 8 \frac{9 e^{\frac{9 x}{8}}}{8} + e^{\frac{9 x}{8}} (2 x) + e^{\frac{9 x}{8}} \cdot - 8$

Step 4.3.5.2

Cancel the common factors.

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Step 4.3.5.2.1

Factor $8$ out of $8$ .

$\frac{x^{2} (9 e^{\frac{9 x}{8}})}{8} + \frac{8 (x (- 9 e^{\frac{9 x}{8}}))}{8 (1)} + 8 \frac{9 e^{\frac{9 x}{8}}}{8} + e^{\frac{9 x}{8}} (2 x) + e^{\frac{9 x}{8}} \cdot - 8$

Step 4.3.5.2.2

Cancel the common factor.

$\frac{x^{2} (9 e^{\frac{9 x}{8}})}{8} + \frac{8 (x (- 9 e^{\frac{9 x}{8}}))}{8 \cdot 1} + 8 \frac{9 e^{\frac{9 x}{8}}}{8} + e^{\frac{9 x}{8}} (2 x) + e^{\frac{9 x}{8}} \cdot - 8$

Step 4.3.5.2.3

Rewrite the expression.

$\frac{x^{2} (9 e^{\frac{9 x}{8}})}{8} + \frac{x (- 9 e^{\frac{9 x}{8}})}{1} + 8 \frac{9 e^{\frac{9 x}{8}}}{8} + e^{\frac{9 x}{8}} (2 x) + e^{\frac{9 x}{8}} \cdot - 8$

Step 4.3.5.2.4

Divide $x (- 9 e^{\frac{9 x}{8}})$ by $1$ .

$\frac{x^{2} (9 e^{\frac{9 x}{8}})}{8} + x (- 9 e^{\frac{9 x}{8}}) + 8 \frac{9 e^{\frac{9 x}{8}}}{8} + e^{\frac{9 x}{8}} (2 x) + e^{\frac{9 x}{8}} \cdot - 8$

Step 4.3.6

Combine $8$ and $\frac{9 e^{\frac{9 x}{8}}}{8}$ .

$\frac{x^{2} (9 e^{\frac{9 x}{8}})}{8} + x (- 9 e^{\frac{9 x}{8}}) + \frac{8 (9 e^{\frac{9 x}{8}})}{8} + e^{\frac{9 x}{8}} (2 x) + e^{\frac{9 x}{8}} \cdot - 8$

Step 4.3.7

Multiply $9$ by $8$ .

$\frac{x^{2} (9 e^{\frac{9 x}{8}})}{8} + x (- 9 e^{\frac{9 x}{8}}) + \frac{72 e^{\frac{9 x}{8}}}{8} + e^{\frac{9 x}{8}} (2 x) + e^{\frac{9 x}{8}} \cdot - 8$

Step 4.3.8

Cancel the common factor of $72$ and $8$ .

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Step 4.3.8.1

Factor $8$ out of $72 e^{\frac{9 x}{8}}$ .

$\frac{x^{2} (9 e^{\frac{9 x}{8}})}{8} + x (- 9 e^{\frac{9 x}{8}}) + \frac{8 (9 e^{\frac{9 x}{8}})}{8} + e^{\frac{9 x}{8}} (2 x) + e^{\frac{9 x}{8}} \cdot - 8$

Step 4.3.8.2

Cancel the common factors.

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Step 4.3.8.2.1

Factor $8$ out of $8$ .

$\frac{x^{2} (9 e^{\frac{9 x}{8}})}{8} + x (- 9 e^{\frac{9 x}{8}}) + \frac{8 (9 e^{\frac{9 x}{8}})}{8 (1)} + e^{\frac{9 x}{8}} (2 x) + e^{\frac{9 x}{8}} \cdot - 8$

Step 4.3.8.2.2

Cancel the common factor.

$\frac{x^{2} (9 e^{\frac{9 x}{8}})}{8} + x (- 9 e^{\frac{9 x}{8}}) + \frac{8 (9 e^{\frac{9 x}{8}})}{8 \cdot 1} + e^{\frac{9 x}{8}} (2 x) + e^{\frac{9 x}{8}} \cdot - 8$

Step 4.3.8.2.3

Rewrite the expression.

$\frac{x^{2} (9 e^{\frac{9 x}{8}})}{8} + x (- 9 e^{\frac{9 x}{8}}) + \frac{9 e^{\frac{9 x}{8}}}{1} + e^{\frac{9 x}{8}} (2 x) + e^{\frac{9 x}{8}} \cdot - 8$

Step 4.3.8.2.4

Divide $9 e^{\frac{9 x}{8}}$ by $1$ .

$\frac{x^{2} (9 e^{\frac{9 x}{8}})}{8} + x (- 9 e^{\frac{9 x}{8}}) + 9 e^{\frac{9 x}{8}} + e^{\frac{9 x}{8}} (2 x) + e^{\frac{9 x}{8}} \cdot - 8$

Step 4.3.9

Move $- 8$ to the left of $e^{\frac{9 x}{8}}$ .

$\frac{x^{2} (9 e^{\frac{9 x}{8}})}{8} + x (- 9 e^{\frac{9 x}{8}}) + 9 e^{\frac{9 x}{8}} + e^{\frac{9 x}{8}} (2 x) - 8 \cdot e^{\frac{9 x}{8}}$

Step 4.3.10

Move $e^{\frac{9 x}{8}}$ .

$\frac{x^{2} (9 e^{\frac{9 x}{8}})}{8} - 9 \cdot x e^{\frac{9 x}{8}} + 2 x e^{\frac{9 x}{8}} + 9 e^{\frac{9 x}{8}} - 8 e^{\frac{9 x}{8}}$

Step 4.3.11

Add $- 9 x e^{\frac{9 x}{8}}$ and $2 x e^{\frac{9 x}{8}}$ .

$\frac{x^{2} (9 e^{\frac{9 x}{8}})}{8} - 7 x e^{\frac{9 x}{8}} + 9 e^{\frac{9 x}{8}} - 8 e^{\frac{9 x}{8}}$

Step 4.3.12

Subtract $8 e^{\frac{9 x}{8}}$ from $9 e^{\frac{9 x}{8}}$ .

$\frac{x^{2} (9 e^{\frac{9 x}{8}})}{8} - 7 x e^{\frac{9 x}{8}} + e^{\frac{9 x}{8}}$

Step 4.4

Reorder terms.

$\frac{x^{2} (9 e^{\frac{9 x}{8}})}{8} - 7 e^{\frac{9 x}{8}} x + e^{\frac{9 x}{8}}$

Step 4.5

Simplify each term.

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Step 4.5.1

Simplify the numerator.

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Step 4.5.1.1

Rewrite.

$\frac{x^{2} (9 e^{\frac{9 x}{8}})}{8} - 7 e^{\frac{9 x}{8}} x + e^{\frac{9 x}{8}}$

Step 4.5.1.2

Remove unnecessary parentheses.

$\frac{x^{2} \cdot 9 e^{\frac{9 x}{8}}}{8} - 7 e^{\frac{9 x}{8}} x + e^{\frac{9 x}{8}}$

Step 4.5.2

Move $9$ to the left of $x^{2}$ .

$\frac{9 x^{2} e^{\frac{9 x}{8}}}{8} - 7 e^{\frac{9 x}{8}} x + e^{\frac{9 x}{8}}$

Step 4.6

Reorder factors in $\frac{9 x^{2} e^{\frac{9 x}{8}}}{8} - 7 e^{\frac{9 x}{8}} x + e^{\frac{9 x}{8}}$ .

$\frac{9 x^{2} e^{\frac{9 x}{8}}}{8} - 7 x e^{\frac{9 x}{8}} + e^{\frac{9 x}{8}}$