Calculus Examples

$\frac{- 5 x^{5} - 3 x^{4} - 4 x^{3}}{x^{4}}$

Step 1

Factor $x^{3}$ out of $- 5 x^{5} - 3 x^{4} - 4 x^{3}$ .

Tap for more steps...

Step 1.1

Factor $x^{3}$ out of $- 5 x^{5}$ .

$\frac{d}{d x} [\frac{x^{3} (- 5 x^{2}) - 3 x^{4} - 4 x^{3}}{x^{4}}]$

Step 1.2

Factor $x^{3}$ out of $- 3 x^{4}$ .

$\frac{d}{d x} [\frac{x^{3} (- 5 x^{2}) + x^{3} (- 3 x) - 4 x^{3}}{x^{4}}]$

Step 1.3

Factor $x^{3}$ out of $- 4 x^{3}$ .

$\frac{d}{d x} [\frac{x^{3} (- 5 x^{2}) + x^{3} (- 3 x) + x^{3} \cdot - 4}{x^{4}}]$

Step 1.4

Factor $x^{3}$ out of $x^{3} (- 5 x^{2}) + x^{3} (- 3 x)$ .

$\frac{d}{d x} [\frac{x^{3} (- 5 x^{2} - 3 x) + x^{3} \cdot - 4}{x^{4}}]$

Step 1.5

Factor $x^{3}$ out of $x^{3} (- 5 x^{2} - 3 x) + x^{3} \cdot - 4$ .

$\frac{d}{d x} [\frac{x^{3} (- 5 x^{2} - 3 x - 4)}{x^{4}}]$

Step 2

Cancel the common factors.

Tap for more steps...

Step 2.1

Factor $x^{3}$ out of $x^{4}$ .

$\frac{d}{d x} [\frac{x^{3} (- 5 x^{2} - 3 x - 4)}{x^{3} x}]$

Step 2.2

Cancel the common factor.

$\frac{d}{d x} [\frac{x^{3} (- 5 x^{2} - 3 x - 4)}{x^{3} x}]$

Step 2.3

Rewrite the expression.

$\frac{d}{d x} [\frac{- 5 x^{2} - 3 x - 4}{x}]$

Step 3

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 $f (x) = - 5 x^{2} - 3 x - 4$ and $g (x) = x$ .

$\frac{x \frac{d}{d x} [- 5 x^{2} - 3 x - 4] - (- 5 x^{2} - 3 x - 4) \frac{d}{d x} [x]}{x^{2}}$

Step 4

Differentiate.

Tap for more steps...

Step 4.1

By the Sum Rule, the derivative of $- 5 x^{2} - 3 x - 4$ with respect to $x$ is $\frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [- 4]$ .

$\frac{x (\frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [- 4]) - (- 5 x^{2} - 3 x - 4) \frac{d}{d x} [x]}{x^{2}}$

Step 4.2

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

$\frac{x (- 5 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [- 4]) - (- 5 x^{2} - 3 x - 4) \frac{d}{d x} [x]}{x^{2}}$

Step 4.3

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

$\frac{x (- 5 (2 x) + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [- 4]) - (- 5 x^{2} - 3 x - 4) \frac{d}{d x} [x]}{x^{2}}$

Step 4.4

Multiply $2$ by $- 5$ .

$\frac{x (- 10 x + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [- 4]) - (- 5 x^{2} - 3 x - 4) \frac{d}{d x} [x]}{x^{2}}$

Step 4.5

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x$ with respect to $x$ is $- 3 \frac{d}{d x} [x]$ .

$\frac{x (- 10 x - 3 \frac{d}{d x} [x] + \frac{d}{d x} [- 4]) - (- 5 x^{2} - 3 x - 4) \frac{d}{d x} [x]}{x^{2}}$

Step 4.6

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

$\frac{x (- 10 x - 3 \cdot 1 + \frac{d}{d x} [- 4]) - (- 5 x^{2} - 3 x - 4) \frac{d}{d x} [x]}{x^{2}}$

Step 4.7

Multiply $- 3$ by $1$ .

$\frac{x (- 10 x - 3 + \frac{d}{d x} [- 4]) - (- 5 x^{2} - 3 x - 4) \frac{d}{d x} [x]}{x^{2}}$

Step 4.8

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$\frac{x (- 10 x - 3 + 0) - (- 5 x^{2} - 3 x - 4) \frac{d}{d x} [x]}{x^{2}}$

Step 4.9

Add $- 10 x - 3$ and $0$ .

$\frac{x (- 10 x - 3) - (- 5 x^{2} - 3 x - 4) \frac{d}{d x} [x]}{x^{2}}$

Step 4.10

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

$\frac{x (- 10 x - 3) - (- 5 x^{2} - 3 x - 4) \cdot 1}{x^{2}}$

Step 4.11

Multiply $- 1$ by $1$ .

$\frac{x (- 10 x - 3) - (- 5 x^{2} - 3 x - 4)}{x^{2}}$

Step 5

Simplify.

Tap for more steps...

Step 5.1

Apply the distributive property.

$\frac{x (- 10 x) + x \cdot - 3 - (- 5 x^{2} - 3 x - 4)}{x^{2}}$

Step 5.2

Apply the distributive property.

$\frac{x (- 10 x) + x \cdot - 3 - (- 5 x^{2}) - (- 3 x) - - 4}{x^{2}}$

Step 5.3

Simplify the numerator.

Tap for more steps...

Step 5.3.1

Simplify each term.

Tap for more steps...

Step 5.3.1.1

Rewrite using the commutative property of multiplication.

$\frac{- 10 x \cdot x + x \cdot - 3 - (- 5 x^{2}) - (- 3 x) - - 4}{x^{2}}$

Step 5.3.1.2

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 5.3.1.2.1

Move $x$ .

$\frac{- 10 (x \cdot x) + x \cdot - 3 - (- 5 x^{2}) - (- 3 x) - - 4}{x^{2}}$

Step 5.3.1.2.2

Multiply $x$ by $x$ .

$\frac{- 10 x^{2} + x \cdot - 3 - (- 5 x^{2}) - (- 3 x) - - 4}{x^{2}}$

Step 5.3.1.3

Move $- 3$ to the left of $x$ .

$\frac{- 10 x^{2} - 3 \cdot x - (- 5 x^{2}) - (- 3 x) - - 4}{x^{2}}$

Step 5.3.1.4

Multiply $- 5$ by $- 1$ .

$\frac{- 10 x^{2} - 3 x + 5 x^{2} - (- 3 x) - - 4}{x^{2}}$

Step 5.3.1.5

Multiply $- 3$ by $- 1$ .

$\frac{- 10 x^{2} - 3 x + 5 x^{2} + 3 x - - 4}{x^{2}}$

Step 5.3.1.6

Multiply $- 1$ by $- 4$ .

$\frac{- 10 x^{2} - 3 x + 5 x^{2} + 3 x + 4}{x^{2}}$

Step 5.3.2

Combine the opposite terms in $- 10 x^{2} - 3 x + 5 x^{2} + 3 x + 4$ .

Tap for more steps...

Step 5.3.2.1

Add $- 3 x$ and $3 x$ .

$\frac{- 10 x^{2} + 5 x^{2} + 0 + 4}{x^{2}}$

Step 5.3.2.2

Add $- 10 x^{2} + 5 x^{2}$ and $0$ .

$\frac{- 10 x^{2} + 5 x^{2} + 4}{x^{2}}$

Step 5.3.3

Add $- 10 x^{2}$ and $5 x^{2}$ .

$\frac{- 5 x^{2} + 4}{x^{2}}$

Step 5.4

Factor $- 1$ out of $- 5 x^{2}$ .

$\frac{- (5 x^{2}) + 4}{x^{2}}$

Step 5.5

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

$\frac{- (5 x^{2}) - 1 (- 4)}{x^{2}}$

Step 5.6

Factor $- 1$ out of $- (5 x^{2}) - 1 (- 4)$ .

$\frac{- (5 x^{2} - 4)}{x^{2}}$

Step 5.7

Rewrite $- (5 x^{2} - 4)$ as $- 1 (5 x^{2} - 4)$ .

$\frac{- 1 (5 x^{2} - 4)}{x^{2}}$

Step 5.8

Move the negative in front of the fraction.

$- \frac{5 x^{2} - 4}{x^{2}}$