Basic Math Examples

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

Step 1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{2} (4 + x)$ with respect to $x$ is $5 \frac{d}{d x} [x^{2} (4 + x)]$ .

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

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{2}$ and $g (x) = 4 + x$ .

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

Move $2$ to the left of $4 + x$ .

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

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Apply the distributive property.

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

Step 4.4

Combine terms.

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Step 4.4.1

Multiply $2$ by $4$ .

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

Step 4.4.2

Multiply $8$ by $5$ .

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

Step 4.4.3

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

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

Step 4.4.4

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

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

Step 4.4.5

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

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

Step 4.4.6

Add $1$ and $1$ .

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

Step 4.4.7

Multiply $2$ by $5$ .

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

Step 4.4.8

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

$15 x^{2} + 40 x$