Find the Derivative - d/dx d/(dx)((x^2+3)/(x^3-2))

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

Step 1

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

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

Step 2

Differentiate.

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

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

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

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

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

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

Simplify the expression.

Add $2 x$ and $0$ .

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

Move $2$ to the left of $x^{3} - 2$ .

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

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

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

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

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

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

$\frac{2 (x^{3} - 2) x - (x^{2} + 3) (3 x^{2} + 0)}{{(x^{3} - 2)}^{2}}$

Simplify the expression.

Add $3 x^{2}$ and $0$ .

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

Multiply $3$ by $- 1$ .

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

Step 3

Simplify.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify the numerator.

Simplify each term.

Multiply $x^{3}$ by $x$ by adding the exponents.

Move $x$ .

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

Multiply $x$ by $x^{3}$ .

Raise $x$ to the power of $1$ .

$\frac{2 (x^{1} x^{3}) + 2 \cdot - 2 x - 3 x^{2} x^{2} - 3 \cdot 3 x^{2}}{{(x^{3} - 2)}^{2}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2 x^{1 + 3} + 2 \cdot - 2 x - 3 x^{2} x^{2} - 3 \cdot 3 x^{2}}{{(x^{3} - 2)}^{2}}$

Add $1$ and $3$ .

$\frac{2 x^{4} + 2 \cdot - 2 x - 3 x^{2} x^{2} - 3 \cdot 3 x^{2}}{{(x^{3} - 2)}^{2}}$

Multiply $2$ by $- 2$ .

$\frac{2 x^{4} - 4 x - 3 x^{2} x^{2} - 3 \cdot 3 x^{2}}{{(x^{3} - 2)}^{2}}$

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$\frac{2 x^{4} - 4 x - 3 (x^{2} x^{2}) - 3 \cdot 3 x^{2}}{{(x^{3} - 2)}^{2}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2 x^{4} - 4 x - 3 x^{2 + 2} - 3 \cdot 3 x^{2}}{{(x^{3} - 2)}^{2}}$

Add $2$ and $2$ .

$\frac{2 x^{4} - 4 x - 3 x^{4} - 3 \cdot 3 x^{2}}{{(x^{3} - 2)}^{2}}$

Multiply $- 3$ by $3$ .

$\frac{2 x^{4} - 4 x - 3 x^{4} - 9 x^{2}}{{(x^{3} - 2)}^{2}}$

Subtract $3 x^{4}$ from $2 x^{4}$ .

$\frac{- x^{4} - 4 x - 9 x^{2}}{{(x^{3} - 2)}^{2}}$

Reorder terms.

$\frac{- x^{4} - 9 x^{2} - 4 x}{{(x^{3} - 2)}^{2}}$

Factor $x$ out of $- x^{4} - 9 x^{2} - 4 x$ .

Factor $x$ out of $- x^{4}$ .

$\frac{x (- x^{3}) - 9 x^{2} - 4 x}{{(x^{3} - 2)}^{2}}$

Factor $x$ out of $- 9 x^{2}$ .

$\frac{x (- x^{3}) + x (- 9 x) - 4 x}{{(x^{3} - 2)}^{2}}$

Factor $x$ out of $- 4 x$ .

$\frac{x (- x^{3}) + x (- 9 x) + x \cdot - 4}{{(x^{3} - 2)}^{2}}$

Factor $x$ out of $x (- x^{3}) + x (- 9 x)$ .

$\frac{x (- x^{3} - 9 x) + x \cdot - 4}{{(x^{3} - 2)}^{2}}$

Factor $x$ out of $x (- x^{3} - 9 x) + x \cdot - 4$ .

$\frac{x (- x^{3} - 9 x - 4)}{{(x^{3} - 2)}^{2}}$

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

$\frac{x (- (x^{3}) - 9 x - 4)}{{(x^{3} - 2)}^{2}}$

Factor $- 1$ out of $- 9 x$ .

$\frac{x (- (x^{3}) - (9 x) - 4)}{{(x^{3} - 2)}^{2}}$

Factor $- 1$ out of $- (x^{3}) - (9 x)$ .

$\frac{x (- (x^{3} + 9 x) - 4)}{{(x^{3} - 2)}^{2}}$

Rewrite $- 4$ as $- 1 (4)$ .

$\frac{x (- (x^{3} + 9 x) - 1 (4))}{{(x^{3} - 2)}^{2}}$

Factor $- 1$ out of $- (x^{3} + 9 x) - 1 (4)$ .

$\frac{x (- (x^{3} + 9 x + 4))}{{(x^{3} - 2)}^{2}}$

Rewrite $- (x^{3} + 9 x + 4)$ as $- 1 (x^{3} + 9 x + 4)$ .

$\frac{x (- 1 (x^{3} + 9 x + 4))}{{(x^{3} - 2)}^{2}}$

Move the negative in front of the fraction.

$- \frac{x (x^{3} + 9 x + 4)}{{(x^{3} - 2)}^{2}}$

Find the Derivative - d/dx d/(dx)((x^2+3)/(x^3-2))

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