Calculus Examples

$y = (- 6 + x - 6 x^{2}) (- 9 - 9 x)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} ((- 6 + x - 6 x^{2}) (- 9 - 9 x))$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

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 $f (x) = - 6 + x - 6 x^{2}$ and $g (x) = - 9 - 9 x$ .

$(- 6 + x - 6 x^{2}) \frac{d}{d x} [- 9 - 9 x] + (- 9 - 9 x) \frac{d}{d x} [- 6 + x - 6 x^{2}]$

Step 3.2

Differentiate.

Tap for more steps...

Step 3.2.1

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

$(- 6 + x - 6 x^{2}) (\frac{d}{d x} [- 9] + \frac{d}{d x} [- 9 x]) + (- 9 - 9 x) \frac{d}{d x} [- 6 + x - 6 x^{2}]$

Step 3.2.2

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9$ with respect to $x$ is $0$ .

$(- 6 + x - 6 x^{2}) (0 + \frac{d}{d x} [- 9 x]) + (- 9 - 9 x) \frac{d}{d x} [- 6 + x - 6 x^{2}]$

Step 3.2.3

Add $0$ and $\frac{d}{d x} [- 9 x]$ .

$(- 6 + x - 6 x^{2}) \frac{d}{d x} [- 9 x] + (- 9 - 9 x) \frac{d}{d x} [- 6 + x - 6 x^{2}]$

Step 3.2.4

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

$(- 6 + x - 6 x^{2}) (- 9 \frac{d}{d x} [x]) + (- 9 - 9 x) \frac{d}{d x} [- 6 + x - 6 x^{2}]$

Step 3.2.5

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

$(- 6 + x - 6 x^{2}) (- 9 \cdot 1) + (- 9 - 9 x) \frac{d}{d x} [- 6 + x - 6 x^{2}]$

Step 3.2.6

Simplify the expression.

Tap for more steps...

Step 3.2.6.1

Multiply $- 9$ by $1$ .

$(- 6 + x - 6 x^{2}) \cdot - 9 + (- 9 - 9 x) \frac{d}{d x} [- 6 + x - 6 x^{2}]$

Step 3.2.6.2

Move $- 9$ to the left of $- 6 + x - 6 x^{2}$ .

$- 9 \cdot (- 6 + x - 6 x^{2}) + (- 9 - 9 x) \frac{d}{d x} [- 6 + x - 6 x^{2}]$

Step 3.2.7

By the Sum Rule, the derivative of $- 6 + x - 6 x^{2}$ with respect to $x$ is $\frac{d}{d x} [- 6] + \frac{d}{d x} [x] + \frac{d}{d x} [- 6 x^{2}]$ .

$- 9 (- 6 + x - 6 x^{2}) + (- 9 - 9 x) (\frac{d}{d x} [- 6] + \frac{d}{d x} [x] + \frac{d}{d x} [- 6 x^{2}])$

Step 3.2.8

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$- 9 (- 6 + x - 6 x^{2}) + (- 9 - 9 x) (0 + \frac{d}{d x} [x] + \frac{d}{d x} [- 6 x^{2}])$

Step 3.2.9

Add $0$ and $\frac{d}{d x} [x]$ .

$- 9 (- 6 + x - 6 x^{2}) + (- 9 - 9 x) (\frac{d}{d x} [x] + \frac{d}{d x} [- 6 x^{2}])$

Step 3.2.10

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

$- 9 (- 6 + x - 6 x^{2}) + (- 9 - 9 x) (1 + \frac{d}{d x} [- 6 x^{2}])$

Step 3.2.11

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x^{2}$ with respect to $x$ is $- 6 \frac{d}{d x} [x^{2}]$ .

$- 9 (- 6 + x - 6 x^{2}) + (- 9 - 9 x) (1 - 6 \frac{d}{d x} [x^{2}])$

Step 3.2.12

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

$- 9 (- 6 + x - 6 x^{2}) + (- 9 - 9 x) (1 - 6 (2 x))$

Step 3.2.13

Multiply $2$ by $- 6$ .

$- 9 (- 6 + x - 6 x^{2}) + (- 9 - 9 x) (1 - 12 x)$

Step 3.3

Simplify.

Tap for more steps...

Step 3.3.1

Apply the distributive property.

$- 9 \cdot - 6 - 9 x - 9 (- 6 x^{2}) + (- 9 - 9 x) (1 - 12 x)$

Step 3.3.2

Apply the distributive property.

$- 9 \cdot - 6 - 9 x - 9 (- 6 x^{2}) + (- 9 - 9 x) \cdot 1 + (- 9 - 9 x) (- 12 x)$

Step 3.3.3

Apply the distributive property.

$- 9 \cdot - 6 - 9 x - 9 (- 6 x^{2}) - 9 \cdot 1 - 9 x \cdot 1 + (- 9 - 9 x) (- 12 x)$

Step 3.3.4

Apply the distributive property.

$- 9 \cdot - 6 - 9 x - 9 (- 6 x^{2}) - 9 \cdot 1 - 9 x \cdot 1 - 9 (- 12 x) - 9 x (- 12 x)$

Step 3.3.5

Combine terms.

Tap for more steps...

Step 3.3.5.1

Multiply $- 9$ by $- 6$ .

$54 - 9 x - 9 (- 6 x^{2}) - 9 \cdot 1 - 9 x \cdot 1 - 9 (- 12 x) - 9 x (- 12 x)$

Step 3.3.5.2

Multiply $- 6$ by $- 9$ .

$54 - 9 x + 54 x^{2} - 9 \cdot 1 - 9 x \cdot 1 - 9 (- 12 x) - 9 x (- 12 x)$

Step 3.3.5.3

Multiply $- 9$ by $1$ .

$54 - 9 x + 54 x^{2} - 9 - 9 x \cdot 1 - 9 (- 12 x) - 9 x (- 12 x)$

Step 3.3.5.4

Multiply $- 9$ by $1$ .

$54 - 9 x + 54 x^{2} - 9 - 9 x - 9 (- 12 x) - 9 x (- 12 x)$

Step 3.3.5.5

Multiply $- 12$ by $- 9$ .

$54 - 9 x + 54 x^{2} - 9 - 9 x + 108 x - 9 x (- 12 x)$

Step 3.3.5.6

Multiply $- 12$ by $- 9$ .

$54 - 9 x + 54 x^{2} - 9 - 9 x + 108 x + 108 x \cdot x$

Step 3.3.5.7

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

$54 - 9 x + 54 x^{2} - 9 - 9 x + 108 x + 108 (x^{1} x)$

Step 3.3.5.8

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

$54 - 9 x + 54 x^{2} - 9 - 9 x + 108 x + 108 (x^{1} x^{1})$

Step 3.3.5.9

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

$54 - 9 x + 54 x^{2} - 9 - 9 x + 108 x + 108 x^{1 + 1}$

Step 3.3.5.10

Add $1$ and $1$ .

$54 - 9 x + 54 x^{2} - 9 - 9 x + 108 x + 108 x^{2}$

Step 3.3.5.11

Add $- 9 x$ and $108 x$ .

$54 - 9 x + 54 x^{2} - 9 + 99 x + 108 x^{2}$

Step 3.3.5.12

Subtract $9$ from $54$ .

$- 9 x + 54 x^{2} + 45 + 99 x + 108 x^{2}$

Step 3.3.5.13

Add $- 9 x$ and $99 x$ .

$90 x + 54 x^{2} + 45 + 108 x^{2}$

Step 3.3.5.14

Add $54 x^{2}$ and $108 x^{2}$ .

$90 x + 162 x^{2} + 45$

Step 3.3.6

Reorder terms.

$162 x^{2} + 90 x + 45$

Step 4

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

$y' = 162 x^{2} + 90 x + 45$

Step 5

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

$\frac{d y}{d x} = 162 x^{2} + 90 x + 45$