Calculus Examples

$\frac{3 x e^{x^{2}} + 8}{y}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d y} [\frac{f (y)}{g (y)}]$ is $\frac{g (y) \frac{d}{d y} [f (y)] - f (y) \frac{d}{d y} [g (y)]}{{g (y)}^{2}}$ where $f (y) = 3 x e^{x^{2}} + 8$ and $g (y) = y$ .

$\frac{y \frac{d}{d y} [3 x e^{x^{2}} + 8] - (3 x e^{x^{2}} + 8) \frac{d}{d y} [y]}{y^{2}}$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $3 x e^{x^{2}} + 8$ with respect to $y$ is $\frac{d}{d y} [3 x e^{x^{2}}] + \frac{d}{d y} [8]$ .

$\frac{y (\frac{d}{d y} [3 x e^{x^{2}}] + \frac{d}{d y} [8]) - (3 x e^{x^{2}} + 8) \frac{d}{d y} [y]}{y^{2}}$

Step 2.2

Since $3 x e^{x^{2}}$ is constant with respect to $y$ , the derivative of $3 x e^{x^{2}}$ with respect to $y$ is $0$ .

$\frac{y (0 + \frac{d}{d y} [8]) - (3 x e^{x^{2}} + 8) \frac{d}{d y} [y]}{y^{2}}$

Step 2.3

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

$\frac{y (0 + 0) - (3 x e^{x^{2}} + 8) \frac{d}{d y} [y]}{y^{2}}$

Step 2.4

Add $0$ and $0$ .

$\frac{y \cdot 0 - (3 x e^{x^{2}} + 8) \frac{d}{d y} [y]}{y^{2}}$

Step 2.5

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

$\frac{y \cdot 0 - (3 x e^{x^{2}} + 8) \cdot 1}{y^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{y \cdot 0 + (- (3 x e^{x^{2}}) - 1 \cdot 8) \cdot 1}{y^{2}}$

Step 3.2

Apply the distributive property.

$\frac{y \cdot 0 - (3 x e^{x^{2}}) \cdot 1 - 1 \cdot 8 \cdot 1}{y^{2}}$

Step 3.3

Simplify the numerator.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Multiply $y$ by $0$ .

$\frac{0 - (3 x e^{x^{2}}) \cdot 1 - 1 \cdot 8 \cdot 1}{y^{2}}$

Step 3.3.1.2

Multiply $3$ by $- 1$ .

$\frac{0 - 3 (x e^{x^{2}}) \cdot 1 - 1 \cdot 8 \cdot 1}{y^{2}}$

Step 3.3.1.3

Multiply $- 3$ by $1$ .

$\frac{0 - 3 x e^{x^{2}} - 1 \cdot 8 \cdot 1}{y^{2}}$

Step 3.3.1.4

Multiply $- 1 \cdot 8 \cdot 1$ .

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Step 3.3.1.4.1

Multiply $- 1$ by $8$ .

$\frac{0 - 3 x e^{x^{2}} - 8 \cdot 1}{y^{2}}$

Step 3.3.1.4.2

Multiply $- 8$ by $1$ .

$\frac{0 - 3 x e^{x^{2}} - 8}{y^{2}}$

Step 3.3.2

Subtract $3 x e^{x^{2}}$ from $0$ .

$\frac{- 3 x e^{x^{2}} - 8}{y^{2}}$

Step 3.4

Reorder terms.

$\frac{- 3 e^{x^{2}} x - 8}{y^{2}}$

Step 3.5

Factor $- 1$ out of $- 3 e^{x^{2}} x$ .

$\frac{- (3 e^{x^{2}} x) - 8}{y^{2}}$

Step 3.6

Rewrite $- 8$ as $- 1 (8)$ .

$\frac{- (3 e^{x^{2}} x) - 1 (8)}{y^{2}}$

Step 3.7

Factor $- 1$ out of $- (3 e^{x^{2}} x) - 1 (8)$ .

$\frac{- (3 e^{x^{2}} x + 8)}{y^{2}}$

Step 3.8

Rewrite $- (3 e^{x^{2}} x + 8)$ as $- 1 (3 e^{x^{2}} x + 8)$ .

$\frac{- 1 (3 e^{x^{2}} x + 8)}{y^{2}}$

Step 3.9

Move the negative in front of the fraction.

$- \frac{3 e^{x^{2}} x + 8}{y^{2}}$

Step 3.10

Reorder factors in $- \frac{3 e^{x^{2}} x + 8}{y^{2}}$ .

$- \frac{3 x e^{x^{2}} + 8}{y^{2}}$