Calculus Examples

$y = (9 x^{2} - 18 x + 18) e^{x}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d y} (y) = \frac{d}{d y} ((9 x^{2} - 18 x + 18) e^{x})$

Step 2

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

$1$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the Product Rule which states that $\frac{d}{d y} [f (y) g (y)]$ is $f (y) \frac{d}{d y} [g (y)] + g (y) \frac{d}{d y} [f (y)]$ where $f (y) = 9 x^{2} - 18 x + 18$ and $g (y) = e^{x}$ .

$(9 x^{2} - 18 x + 18) \frac{d}{d y} [e^{x}] + e^{x} \frac{d}{d y} [9 x^{2} - 18 x + 18]$

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = e^{y}$ and $g (y) = x$ .

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Step 3.2.1

To apply the Chain Rule, set $u_{1}$ as $x$ .

$(9 x^{2} - 18 x + 18) (\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d y} [x]) + e^{x} \frac{d}{d y} [9 x^{2} - 18 x + 18]$

Step 3.2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$(9 x^{2} - 18 x + 18) (e^{u_{1}} \frac{d}{d y} [x]) + e^{x} \frac{d}{d y} [9 x^{2} - 18 x + 18]$

Step 3.2.3

Replace all occurrences of $u_{1}$ with $x$ .

$(9 x^{2} - 18 x + 18) (e^{x} \frac{d}{d y} [x]) + e^{x} \frac{d}{d y} [9 x^{2} - 18 x + 18]$

Step 3.3

Rewrite $\frac{d}{d y} [x]$ as $x'$ .

$(9 x^{2} - 18 x + 18) e^{x} x' + e^{x} \frac{d}{d y} [9 x^{2} - 18 x + 18]$

Step 3.4

Differentiate.

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Step 3.4.1

By the Sum Rule, the derivative of $9 x^{2} - 18 x + 18$ with respect to $y$ is $\frac{d}{d y} [9 x^{2}] + \frac{d}{d y} [- 18 x] + \frac{d}{d y} [18]$ .

$(9 x^{2} - 18 x + 18) e^{x} x' + e^{x} (\frac{d}{d y} [9 x^{2}] + \frac{d}{d y} [- 18 x] + \frac{d}{d y} [18])$

Step 3.4.2

Since $9$ is constant with respect to $y$ , the derivative of $9 x^{2}$ with respect to $y$ is $9 \frac{d}{d y} [x^{2}]$ .

$(9 x^{2} - 18 x + 18) e^{x} x' + e^{x} (9 \frac{d}{d y} [x^{2}] + \frac{d}{d y} [- 18 x] + \frac{d}{d y} [18])$

Step 3.5

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = y^{2}$ and $g (y) = x$ .

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Step 3.5.1

To apply the Chain Rule, set $u_{2}$ as $x$ .

$(9 x^{2} - 18 x + 18) e^{x} x' + e^{x} (9 (\frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d y} [x]) + \frac{d}{d y} [- 18 x] + \frac{d}{d y} [18])$

Step 3.5.2

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

$(9 x^{2} - 18 x + 18) e^{x} x' + e^{x} (9 (2 u_{2} \frac{d}{d y} [x]) + \frac{d}{d y} [- 18 x] + \frac{d}{d y} [18])$

Step 3.5.3

Replace all occurrences of $u_{2}$ with $x$ .

$(9 x^{2} - 18 x + 18) e^{x} x' + e^{x} (9 (2 x \frac{d}{d y} [x]) + \frac{d}{d y} [- 18 x] + \frac{d}{d y} [18])$

Step 3.6

Multiply $2$ by $9$ .

$(9 x^{2} - 18 x + 18) e^{x} x' + e^{x} (18 (x \frac{d}{d y} [x]) + \frac{d}{d y} [- 18 x] + \frac{d}{d y} [18])$

Step 3.7

Rewrite $\frac{d}{d y} [x]$ as $x'$ .

$(9 x^{2} - 18 x + 18) e^{x} x' + e^{x} (18 x x' + \frac{d}{d y} [- 18 x] + \frac{d}{d y} [18])$

Step 3.8

Since $- 18$ is constant with respect to $y$ , the derivative of $- 18 x$ with respect to $y$ is $- 18 \frac{d}{d y} [x]$ .

$(9 x^{2} - 18 x + 18) e^{x} x' + e^{x} (18 x x' - 18 \frac{d}{d y} [x] + \frac{d}{d y} [18])$

Step 3.9

Rewrite $\frac{d}{d y} [x]$ as $x'$ .

$(9 x^{2} - 18 x + 18) e^{x} x' + e^{x} (18 x x' - 18 x' + \frac{d}{d y} [18])$

Step 3.10

Since $18$ is constant with respect to $y$ , the derivative of $18$ with respect to $y$ is $0$ .

$(9 x^{2} - 18 x + 18) e^{x} x' + e^{x} (18 x x' - 18 x' + 0)$

Step 3.11

Add $18 x x' - 18 x'$ and $0$ .

$(9 x^{2} - 18 x + 18) e^{x} x' + e^{x} (18 x x' - 18 x')$

Step 3.12

Simplify.

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Step 3.12.1

Apply the distributive property.

$(9 x^{2} e^{x} - 18 x e^{x} + 18 e^{x}) x' + e^{x} (18 x x' - 18 x')$

Step 3.12.2

Apply the distributive property.

$9 x^{2} e^{x} x' - 18 x e^{x} x' + 18 e^{x} x' + e^{x} (18 x x' - 18 x')$

Step 3.12.3

Apply the distributive property.

$9 x^{2} e^{x} x' - 18 x e^{x} x' + 18 e^{x} x' + e^{x} (18 x x') + e^{x} (- 18 x')$

Step 3.12.4

Combine terms.

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Step 3.12.4.1

Add $- 18 x e^{x} x'$ and $e^{x} (18 x x')$ .

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Step 3.12.4.1.1

Move $e^{x}$ .

$9 x^{2} e^{x} x' + 18 e^{x} x' - 18 x e^{x} x' + 18 x e^{x} x' + e^{x} (- 18 x')$

Step 3.12.4.1.2

Add $- 18 x e^{x} x'$ and $18 x e^{x} x'$ .

$9 x^{2} e^{x} x' + 18 e^{x} x' + 0 + e^{x} (- 18 x')$

Step 3.12.4.2

Add $9 x^{2} e^{x} x' + 18 e^{x} x'$ and $0$ .

$9 x^{2} e^{x} x' + 18 e^{x} x' + e^{x} (- 18 x')$

Step 3.12.4.3

Add $18 e^{x} x'$ and $e^{x} (- 18 x')$ .

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Step 3.12.4.3.1

Reorder $e^{x}$ and $- 18$ .

$9 x^{2} e^{x} x' + 18 e^{x} x' - 18 \cdot e^{x} x'$

Step 3.12.4.3.2

Subtract $18 \cdot e^{x} x'$ from $18 e^{x} x'$ .

$9 x^{2} e^{x} x' + 0$

Step 3.12.4.4

Add $9 x^{2} e^{x} x'$ and $0$ .

$9 x^{2} e^{x} x'$

Step 3.12.5

Reorder the factors of $9 x^{2} e^{x} x'$ .

$9 e^{x} x^{2} x'$

Step 3.12.6

Reorder factors in $9 e^{x} x^{2} x'$ .

$9 x^{2} e^{x} x'$

Step 4

Reform the equation by setting the left side equal to the right side.

$1 = 9 x^{2} e^{x} x'$

Step 5

Solve for $x'$ .

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Step 5.1

Rewrite the equation as $9 x^{2} e^{x} x' = 1$ .

$9 x^{2} e^{x} x' = 1$

Step 5.2

Divide each term in $9 x^{2} e^{x} x' = 1$ by $9 x^{2} e^{x}$ and simplify.

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Step 5.2.1

Divide each term in $9 x^{2} e^{x} x' = 1$ by $9 x^{2} e^{x}$ .

$\frac{9 x^{2} e^{x} x'}{9 x^{2} e^{x}} = \frac{1}{9 x^{2} e^{x}}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $9$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{9 x^{2} e^{x} x'}{9 x^{2} e^{x}} = \frac{1}{9 x^{2} e^{x}}$

Step 5.2.2.1.2

Rewrite the expression.

$\frac{x^{2} e^{x} x'}{x^{2} e^{x}} = \frac{1}{9 x^{2} e^{x}}$

Step 5.2.2.2

Cancel the common factor of $x^{2}$ .

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Step 5.2.2.2.1

Cancel the common factor.

$\frac{x^{2} e^{x} x'}{x^{2} e^{x}} = \frac{1}{9 x^{2} e^{x}}$

Step 5.2.2.2.2

Rewrite the expression.

$\frac{e^{x} x'}{e^{x}} = \frac{1}{9 x^{2} e^{x}}$

Step 5.2.2.3

Cancel the common factor of $e^{x}$ .

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Step 5.2.2.3.1

Cancel the common factor.

$\frac{e^{x} x'}{e^{x}} = \frac{1}{9 x^{2} e^{x}}$

Step 5.2.2.3.2

Divide $x'$ by $1$ .

$x' = \frac{1}{9 x^{2} e^{x}}$

Step 6

Replace $x'$ with $\frac{d x}{d y}$ .

$\frac{d x}{d y} = \frac{1}{9 x^{2} e^{x}}$