Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

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

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

Step 3.2

Differentiate.

Tap for more steps...

Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.2.4

Add $2 x$ and $0$ .

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

Add $1$ and $1$ .

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

Step 3.7

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

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

Step 3.8

Simplify by adding terms.

Tap for more steps...

Step 3.8.1

Multiply $x^{2} - 6$ by $1$ .

$2 x^{2} + x^{2} - 6$

Step 3.8.2

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

$3 x^{2} - 6$

Step 4

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

$y' = 3 x^{2} - 6$

Step 5

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

$\frac{d y}{d x} = 3 x^{2} - 6$