Basic Math Examples

5 x^{2} (4 + x)

$5 x^{2} (4 + x)$

Step 1

Since

5

$5$ is constant with respect to

x

$x$ , the derivative of

5 x^{2} (4 + x)

$5 x^{2} (4 + x)$ with respect to

x

$x$ is

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

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

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

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

Step 2

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) = 4 + x

$g (x) = 4 + x$ .

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

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

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of

4 + x

$4 + x$ with respect to

x

$x$ is

\frac{d}{d x} [4] + \frac{d}{d x} [x]

$\frac{d}{d x} [4] + \frac{d}{d x} [x]$ .

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

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

Step 3.2

Since

4

$4$ is constant with respect to

x

$x$ , the derivative of

4

$4$ with respect to

x

$x$ is

0

$0$ .

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

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

Step 3.3

Add

0

$0$ and

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

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

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

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

Step 3.4

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

$n = 1$ .

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

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

Step 3.5

Multiply

x^{2}

$x^{2}$ by

1

$1$ .

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

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

Step 3.6

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

$n = 2$ .

5 (x^{2} + (4 + x) (2 x))

$5 (x^{2} + (4 + x) (2 x))$

Step 3.7

Move

2

$2$ to the left of

4 + x

$4 + x$ .

5 (x^{2} + 2 \cdot (4 + x) x)

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

5 (x^{2} + 2 \cdot (4 + x) x)

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

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

5 (x^{2} + (2 \cdot 4 + 2 x) x)

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

Step 4.2

Apply the distributive property.

5 (x^{2} + 2 \cdot 4 x + 2 x \cdot x)

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

Step 4.3

Apply the distributive property.

5 x^{2} + 5 (2 \cdot 4 x) + 5 (2 x \cdot x)

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

Step 4.4

Combine terms.

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Step 4.4.1

Multiply

2

$2$ by

4

$4$ .

5 x^{2} + 5 (8 x) + 5 (2 x \cdot x)

$5 x^{2} + 5 (8 x) + 5 (2 x \cdot x)$

Step 4.4.2

Multiply

8

$8$ by

5

$5$ .

5 x^{2} + 40 x + 5 (2 x \cdot x)

$5 x^{2} + 40 x + 5 (2 x \cdot x)$

Step 4.4.3

Raise

x

$x$ to the power of

1

$1$ .

5 x^{2} + 40 x + 5 (2 (x^{1} x))

$5 x^{2} + 40 x + 5 (2 (x^{1} x))$

Step 4.4.4

Raise

x

$x$ to the power of

1

$1$ .

5 x^{2} + 40 x + 5 (2 (x^{1} x^{1}))

$5 x^{2} + 40 x + 5 (2 (x^{1} x^{1}))$

Step 4.4.5

Use the power rule

a^{m} a^{n} = a^{m + n}

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

5 x^{2} + 40 x + 5 (2 x^{1 + 1})

$5 x^{2} + 40 x + 5 (2 x^{1 + 1})$

Step 4.4.6

Add

1

$1$ and

1

$1$ .

5 x^{2} + 40 x + 5 (2 x^{2})

$5 x^{2} + 40 x + 5 (2 x^{2})$

Step 4.4.7

Multiply

2

$2$ by

5

$5$ .

5 x^{2} + 40 x + 10 x^{2}

$5 x^{2} + 40 x + 10 x^{2}$

Step 4.4.8

Add

5 x^{2}

$5 x^{2}$ and

10 x^{2}

$10 x^{2}$ .

15 x^{2} + 40 x

$15 x^{2} + 40 x$

15 x^{2} + 40 x

$15 x^{2} + 40 x$

15 x^{2} + 40 x

$15 x^{2} + 40 x$