Calculus Examples

x^{2} (1 - 8 x)

$x^{2} (1 - 8 x)$

Step 1

Differentiate using the Product Rule which states that

\frac{d}{d x} [f (x) g (x)]

$\frac{d}{d x} [f (x) g (x)]$ is

f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]

$f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where

f (x) = x^{2}

$f (x) = x^{2}$ and

g (x) = 1 - 8 x

$g (x) = 1 - 8 x$ .

x^{2} \frac{d}{d x} [1 - 8 x] + (1 - 8 x) \frac{d}{d x} [x^{2}]

$x^{2} \frac{d}{d x} [1 - 8 x] + (1 - 8 x) \frac{d}{d x} [x^{2}]$

Step 2

Differentiate.

Tap for more steps...

Step 2.1

By the Sum Rule, the derivative of

1 - 8 x

$1 - 8 x$ with respect to

x

$x$ is

\frac{d}{d x} [1] + \frac{d}{d x} [- 8 x]

$\frac{d}{d x} [1] + \frac{d}{d x} [- 8 x]$ .

x^{2} (\frac{d}{d x} [1] + \frac{d}{d x} [- 8 x]) + (1 - 8 x) \frac{d}{d x} [x^{2}]

$x^{2} (\frac{d}{d x} [1] + \frac{d}{d x} [- 8 x]) + (1 - 8 x) \frac{d}{d x} [x^{2}]$

Step 2.2

Since

1

$1$ is constant with respect to

x

$x$ , the derivative of

1

$1$ with respect to

x

$x$ is

0

$0$ .

x^{2} (0 + \frac{d}{d x} [- 8 x]) + (1 - 8 x) \frac{d}{d x} [x^{2}]

$x^{2} (0 + \frac{d}{d x} [- 8 x]) + (1 - 8 x) \frac{d}{d x} [x^{2}]$

Step 2.3

Add

0

$0$ and

\frac{d}{d x} [- 8 x]

$\frac{d}{d x} [- 8 x]$ .

x^{2} \frac{d}{d x} [- 8 x] + (1 - 8 x) \frac{d}{d x} [x^{2}]

$x^{2} \frac{d}{d x} [- 8 x] + (1 - 8 x) \frac{d}{d x} [x^{2}]$

Step 2.4

Since

- 8

$- 8$ is constant with respect to

x

$x$ , the derivative of

- 8 x

$- 8 x$ with respect to

x

$x$ is

- 8 \frac{d}{d x} [x]

$- 8 \frac{d}{d x} [x]$ .

x^{2} (- 8 \frac{d}{d x} [x]) + (1 - 8 x) \frac{d}{d x} [x^{2}]

$x^{2} (- 8 \frac{d}{d x} [x]) + (1 - 8 x) \frac{d}{d x} [x^{2}]$

Step 2.5

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 1

$n = 1$ .

x^{2} (- 8 \cdot 1) + (1 - 8 x) \frac{d}{d x} [x^{2}]

$x^{2} (- 8 \cdot 1) + (1 - 8 x) \frac{d}{d x} [x^{2}]$

Step 2.6

Simplify the expression.

Tap for more steps...

Step 2.6.1

Multiply

- 8

$- 8$ by

1

$1$ .

x^{2} \cdot - 8 + (1 - 8 x) \frac{d}{d x} [x^{2}]

$x^{2} \cdot - 8 + (1 - 8 x) \frac{d}{d x} [x^{2}]$

Step 2.6.2

Move

- 8

$- 8$ to the left of

x^{2}

$x^{2}$ .

- 8 \cdot x^{2} + (1 - 8 x) \frac{d}{d x} [x^{2}]

$- 8 \cdot x^{2} + (1 - 8 x) \frac{d}{d x} [x^{2}]$

- 8 \cdot x^{2} + (1 - 8 x) \frac{d}{d x} [x^{2}]

$- 8 \cdot x^{2} + (1 - 8 x) \frac{d}{d x} [x^{2}]$

Step 2.7

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 2

$n = 2$ .

- 8 x^{2} + (1 - 8 x) (2 x)

$- 8 x^{2} + (1 - 8 x) (2 x)$

Step 2.8

Move

2

$2$ to the left of

1 - 8 x

$1 - 8 x$ .

- 8 x^{2} + 2 \cdot (1 - 8 x) x

$- 8 x^{2} + 2 \cdot (1 - 8 x) x$

- 8 x^{2} + 2 \cdot (1 - 8 x) x

$- 8 x^{2} + 2 \cdot (1 - 8 x) x$

Step 3

Simplify.

Tap for more steps...

Step 3.1

Apply the distributive property.

- 8 x^{2} + (2 \cdot 1 + 2 (- 8 x)) x

$- 8 x^{2} + (2 \cdot 1 + 2 (- 8 x)) x$

Step 3.2

Apply the distributive property.

- 8 x^{2} + 2 \cdot 1 x + 2 (- 8 x) x

$- 8 x^{2} + 2 \cdot 1 x + 2 (- 8 x) x$

Step 3.3

Combine terms.

Tap for more steps...

Step 3.3.1

Multiply

$2$ by

$1$ .

$- 8 x^{2} + 2 x + 2 (- 8 x) x$

Step 3.3.2

Multiply

$- 8$ by

$2$ .

$- 8 x^{2} + 2 x - 16 x \cdot x$

Step 3.3.3

Raise

$x$ to the power of

$1$ .

$- 8 x^{2} + 2 x - 16 (x^{1} x)$

Step 3.3.4

Raise

$x$ to the power of

$1$ .

$- 8 x^{2} + 2 x - 16 (x^{1} x^{1})$

Step 3.3.5

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 8 x^{2} + 2 x - 16 x^{1 + 1}$

Step 3.3.6

Add

$1$ and

$1$ .

$- 8 x^{2} + 2 x - 16 x^{2}$

Step 3.3.7

Subtract

$16 x^{2}$ from

$- 8 x^{2}$ .

$- 24 x^{2} + 2 x$