Calculus Examples

$y = (2 x) (x + 5)$

Step 1

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

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

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 $f (x) = x$ and $g (x) = x + 5$ .

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

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $x + 5$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [5]$ .

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

Step 3.2

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

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

Step 3.3

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$2 (x (1 + 0) + (x + 5) \frac{d}{d x} [x])$

Step 3.4

Simplify the expression.

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Step 3.4.1

Add $1$ and $0$ .

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

Step 3.4.2

Multiply $x$ by $1$ .

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

Step 3.5

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

$2 (x + (x + 5) \cdot 1)$

Step 3.6

Simplify by adding terms.

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Step 3.6.1

Multiply $x + 5$ by $1$ .

$2 (x + x + 5)$

Step 3.6.2

Add $x$ and $x$ .

$2 (2 x + 5)$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$2 (2 x) + 2 \cdot 5$

Step 4.2

Combine terms.

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Step 4.2.1

Multiply $2$ by $2$ .

$4 x + 2 \cdot 5$

Step 4.2.2

Multiply $2$ by $5$ .

$4 x + 10$