Calculus Examples

$y = (x^{2} - 2 x + 2) e^{\frac{5 x}{2}}$

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 $f (x) = x^{2} - 2 x + 2$ and $g (x) = e^{\frac{5 x}{2}}$ .

$(x^{2} - 2 x + 2) \frac{d}{d x} [e^{\frac{5 x}{2}}] + e^{\frac{5 x}{2}} \frac{d}{d x} [x^{2} - 2 x + 2]$

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 $g (x) = \frac{5 x}{2}$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $\frac{5 x}{2}$ .

$(x^{2} - 2 x + 2) (\frac{d}{d u} [e^{u}] \frac{d}{d x} [\frac{5 x}{2}]) + e^{\frac{5 x}{2}} \frac{d}{d x} [x^{2} - 2 x + 2]$

Step 2.2

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

$(x^{2} - 2 x + 2) (e^{u} \frac{d}{d x} [\frac{5 x}{2}]) + e^{\frac{5 x}{2}} \frac{d}{d x} [x^{2} - 2 x + 2]$

Step 2.3

Replace all occurrences of $u$ with $\frac{5 x}{2}$ .

$(x^{2} - 2 x + 2) (e^{\frac{5 x}{2}} \frac{d}{d x} [\frac{5 x}{2}]) + e^{\frac{5 x}{2}} \frac{d}{d x} [x^{2} - 2 x + 2]$

Step 3

Differentiate.

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Step 3.1

Since $\frac{5}{2}$ is constant with respect to $x$ , the derivative of $\frac{5 x}{2}$ with respect to $x$ is $\frac{5}{2} \frac{d}{d x} [x]$ .

$(x^{2} - 2 x + 2) e^{\frac{5 x}{2}} (\frac{5}{2} \frac{d}{d x} [x]) + e^{\frac{5 x}{2}} \frac{d}{d x} [x^{2} - 2 x + 2]$

Step 3.2

Combine $\frac{5}{2}$ and $e^{\frac{5 x}{2}}$ .

$(x^{2} - 2 x + 2) \frac{5 e^{\frac{5 x}{2}}}{2} \frac{d}{d x} [x] + e^{\frac{5 x}{2}} \frac{d}{d x} [x^{2} - 2 x + 2]$

Step 3.3

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

$(x^{2} - 2 x + 2) \frac{5 e^{\frac{5 x}{2}}}{2} \cdot 1 + e^{\frac{5 x}{2}} \frac{d}{d x} [x^{2} - 2 x + 2]$

Step 3.4

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

$(x^{2} - 2 x + 2) \frac{5 e^{\frac{5 x}{2}}}{2} + e^{\frac{5 x}{2}} \frac{d}{d x} [x^{2} - 2 x + 2]$

Step 3.5

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

$(x^{2} - 2 x + 2) \frac{5 e^{\frac{5 x}{2}}}{2} + e^{\frac{5 x}{2}} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [2])$

Step 3.6

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

$(x^{2} - 2 x + 2) \frac{5 e^{\frac{5 x}{2}}}{2} + e^{\frac{5 x}{2}} (2 x + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [2])$

Step 3.7

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

$(x^{2} - 2 x + 2) \frac{5 e^{\frac{5 x}{2}}}{2} + e^{\frac{5 x}{2}} (2 x - 2 \frac{d}{d x} [x] + \frac{d}{d x} [2])$

Step 3.8

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

$(x^{2} - 2 x + 2) \frac{5 e^{\frac{5 x}{2}}}{2} + e^{\frac{5 x}{2}} (2 x - 2 \cdot 1 + \frac{d}{d x} [2])$

Step 3.9

Multiply $- 2$ by $1$ .

$(x^{2} - 2 x + 2) \frac{5 e^{\frac{5 x}{2}}}{2} + e^{\frac{5 x}{2}} (2 x - 2 + \frac{d}{d x} [2])$

Step 3.10

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$(x^{2} - 2 x + 2) \frac{5 e^{\frac{5 x}{2}}}{2} + e^{\frac{5 x}{2}} (2 x - 2 + 0)$

Step 3.11

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

$(x^{2} - 2 x + 2) \frac{5 e^{\frac{5 x}{2}}}{2} + e^{\frac{5 x}{2}} (2 x - 2)$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$x^{2} \frac{5 e^{\frac{5 x}{2}}}{2} - 2 x \frac{5 e^{\frac{5 x}{2}}}{2} + 2 \frac{5 e^{\frac{5 x}{2}}}{2} + e^{\frac{5 x}{2}} (2 x - 2)$

Step 4.2

Apply the distributive property.

$x^{2} \frac{5 e^{\frac{5 x}{2}}}{2} - 2 x \frac{5 e^{\frac{5 x}{2}}}{2} + 2 \frac{5 e^{\frac{5 x}{2}}}{2} + e^{\frac{5 x}{2}} (2 x) + e^{\frac{5 x}{2}} \cdot - 2$

Step 4.3

Combine terms.

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Step 4.3.1

Combine $x^{2}$ and $\frac{5 e^{\frac{5 x}{2}}}{2}$ .

$\frac{x^{2} (5 e^{\frac{5 x}{2}})}{2} - 2 x \frac{5 e^{\frac{5 x}{2}}}{2} + 2 \frac{5 e^{\frac{5 x}{2}}}{2} + e^{\frac{5 x}{2}} (2 x) + e^{\frac{5 x}{2}} \cdot - 2$

Step 4.3.2

Combine $- 2$ and $\frac{5 e^{\frac{5 x}{2}}}{2}$ .

$\frac{x^{2} (5 e^{\frac{5 x}{2}})}{2} + x \frac{- 2 (5 e^{\frac{5 x}{2}})}{2} + 2 \frac{5 e^{\frac{5 x}{2}}}{2} + e^{\frac{5 x}{2}} (2 x) + e^{\frac{5 x}{2}} \cdot - 2$

Step 4.3.3

Multiply $5$ by $- 2$ .

$\frac{x^{2} (5 e^{\frac{5 x}{2}})}{2} + x \frac{- 10 e^{\frac{5 x}{2}}}{2} + 2 \frac{5 e^{\frac{5 x}{2}}}{2} + e^{\frac{5 x}{2}} (2 x) + e^{\frac{5 x}{2}} \cdot - 2$

Step 4.3.4

Combine $x$ and $\frac{- 10 e^{\frac{5 x}{2}}}{2}$ .

$\frac{x^{2} (5 e^{\frac{5 x}{2}})}{2} + \frac{x (- 10 e^{\frac{5 x}{2}})}{2} + 2 \frac{5 e^{\frac{5 x}{2}}}{2} + e^{\frac{5 x}{2}} (2 x) + e^{\frac{5 x}{2}} \cdot - 2$

Step 4.3.5

Cancel the common factor of $- 10$ and $2$ .

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Step 4.3.5.1

Factor $2$ out of $x (- 10 e^{\frac{5 x}{2}})$ .

$\frac{x^{2} (5 e^{\frac{5 x}{2}})}{2} + \frac{2 (x (- 5 e^{\frac{5 x}{2}}))}{2} + 2 \frac{5 e^{\frac{5 x}{2}}}{2} + e^{\frac{5 x}{2}} (2 x) + e^{\frac{5 x}{2}} \cdot - 2$

Step 4.3.5.2

Cancel the common factors.

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Step 4.3.5.2.1

Factor $2$ out of $2$ .

$\frac{x^{2} (5 e^{\frac{5 x}{2}})}{2} + \frac{2 (x (- 5 e^{\frac{5 x}{2}}))}{2 (1)} + 2 \frac{5 e^{\frac{5 x}{2}}}{2} + e^{\frac{5 x}{2}} (2 x) + e^{\frac{5 x}{2}} \cdot - 2$

Step 4.3.5.2.2

Cancel the common factor.

$\frac{x^{2} (5 e^{\frac{5 x}{2}})}{2} + \frac{2 (x (- 5 e^{\frac{5 x}{2}}))}{2 \cdot 1} + 2 \frac{5 e^{\frac{5 x}{2}}}{2} + e^{\frac{5 x}{2}} (2 x) + e^{\frac{5 x}{2}} \cdot - 2$

Step 4.3.5.2.3

Rewrite the expression.

$\frac{x^{2} (5 e^{\frac{5 x}{2}})}{2} + \frac{x (- 5 e^{\frac{5 x}{2}})}{1} + 2 \frac{5 e^{\frac{5 x}{2}}}{2} + e^{\frac{5 x}{2}} (2 x) + e^{\frac{5 x}{2}} \cdot - 2$

Step 4.3.5.2.4

Divide $x (- 5 e^{\frac{5 x}{2}})$ by $1$ .

$\frac{x^{2} (5 e^{\frac{5 x}{2}})}{2} + x (- 5 e^{\frac{5 x}{2}}) + 2 \frac{5 e^{\frac{5 x}{2}}}{2} + e^{\frac{5 x}{2}} (2 x) + e^{\frac{5 x}{2}} \cdot - 2$

Step 4.3.6

Combine $2$ and $\frac{5 e^{\frac{5 x}{2}}}{2}$ .

$\frac{x^{2} (5 e^{\frac{5 x}{2}})}{2} + x (- 5 e^{\frac{5 x}{2}}) + \frac{2 (5 e^{\frac{5 x}{2}})}{2} + e^{\frac{5 x}{2}} (2 x) + e^{\frac{5 x}{2}} \cdot - 2$

Step 4.3.7

Multiply $5$ by $2$ .

$\frac{x^{2} (5 e^{\frac{5 x}{2}})}{2} + x (- 5 e^{\frac{5 x}{2}}) + \frac{10 e^{\frac{5 x}{2}}}{2} + e^{\frac{5 x}{2}} (2 x) + e^{\frac{5 x}{2}} \cdot - 2$

Step 4.3.8

Cancel the common factor of $10$ and $2$ .

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Step 4.3.8.1

Factor $2$ out of $10 e^{\frac{5 x}{2}}$ .

$\frac{x^{2} (5 e^{\frac{5 x}{2}})}{2} + x (- 5 e^{\frac{5 x}{2}}) + \frac{2 (5 e^{\frac{5 x}{2}})}{2} + e^{\frac{5 x}{2}} (2 x) + e^{\frac{5 x}{2}} \cdot - 2$

Step 4.3.8.2

Cancel the common factors.

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Step 4.3.8.2.1

Factor $2$ out of $2$ .

$\frac{x^{2} (5 e^{\frac{5 x}{2}})}{2} + x (- 5 e^{\frac{5 x}{2}}) + \frac{2 (5 e^{\frac{5 x}{2}})}{2 (1)} + e^{\frac{5 x}{2}} (2 x) + e^{\frac{5 x}{2}} \cdot - 2$

Step 4.3.8.2.2

Cancel the common factor.

$\frac{x^{2} (5 e^{\frac{5 x}{2}})}{2} + x (- 5 e^{\frac{5 x}{2}}) + \frac{2 (5 e^{\frac{5 x}{2}})}{2 \cdot 1} + e^{\frac{5 x}{2}} (2 x) + e^{\frac{5 x}{2}} \cdot - 2$

Step 4.3.8.2.3

Rewrite the expression.

$\frac{x^{2} (5 e^{\frac{5 x}{2}})}{2} + x (- 5 e^{\frac{5 x}{2}}) + \frac{5 e^{\frac{5 x}{2}}}{1} + e^{\frac{5 x}{2}} (2 x) + e^{\frac{5 x}{2}} \cdot - 2$

Step 4.3.8.2.4

Divide $5 e^{\frac{5 x}{2}}$ by $1$ .

$\frac{x^{2} (5 e^{\frac{5 x}{2}})}{2} + x (- 5 e^{\frac{5 x}{2}}) + 5 e^{\frac{5 x}{2}} + e^{\frac{5 x}{2}} (2 x) + e^{\frac{5 x}{2}} \cdot - 2$

Step 4.3.9

Move $- 2$ to the left of $e^{\frac{5 x}{2}}$ .

$\frac{x^{2} (5 e^{\frac{5 x}{2}})}{2} + x (- 5 e^{\frac{5 x}{2}}) + 5 e^{\frac{5 x}{2}} + e^{\frac{5 x}{2}} (2 x) - 2 \cdot e^{\frac{5 x}{2}}$

Step 4.3.10

Move $e^{\frac{5 x}{2}}$ .

$\frac{x^{2} (5 e^{\frac{5 x}{2}})}{2} - 5 \cdot x e^{\frac{5 x}{2}} + 2 x e^{\frac{5 x}{2}} + 5 e^{\frac{5 x}{2}} - 2 e^{\frac{5 x}{2}}$

Step 4.3.11

Add $- 5 x e^{\frac{5 x}{2}}$ and $2 x e^{\frac{5 x}{2}}$ .

$\frac{x^{2} (5 e^{\frac{5 x}{2}})}{2} - 3 x e^{\frac{5 x}{2}} + 5 e^{\frac{5 x}{2}} - 2 e^{\frac{5 x}{2}}$

Step 4.3.12

Subtract $2 e^{\frac{5 x}{2}}$ from $5 e^{\frac{5 x}{2}}$ .

$\frac{x^{2} (5 e^{\frac{5 x}{2}})}{2} - 3 x e^{\frac{5 x}{2}} + 3 e^{\frac{5 x}{2}}$

Step 4.4

Reorder terms.

$\frac{x^{2} (5 e^{\frac{5 x}{2}})}{2} - 3 e^{\frac{5 x}{2}} x + 3 e^{\frac{5 x}{2}}$

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.

$\frac{x^{2} (5 e^{\frac{5 x}{2}})}{2} - 3 e^{\frac{5 x}{2}} x + 3 e^{\frac{5 x}{2}}$

Step 4.5.1.2

Remove unnecessary parentheses.

$\frac{x^{2} \cdot 5 e^{\frac{5 x}{2}}}{2} - 3 e^{\frac{5 x}{2}} x + 3 e^{\frac{5 x}{2}}$

Step 4.5.2

Move $5$ to the left of $x^{2}$ .

$\frac{5 x^{2} e^{\frac{5 x}{2}}}{2} - 3 e^{\frac{5 x}{2}} x + 3 e^{\frac{5 x}{2}}$

Step 4.6

Reorder factors in $\frac{5 x^{2} e^{\frac{5 x}{2}}}{2} - 3 e^{\frac{5 x}{2}} x + 3 e^{\frac{5 x}{2}}$ .

$\frac{5 x^{2} e^{\frac{5 x}{2}}}{2} - 3 x e^{\frac{5 x}{2}} + 3 e^{\frac{5 x}{2}}$