Calculus Examples

$y = (3 x - 1) (2 x + 5)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} ((3 x - 1) (2 x + 5))$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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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 $f (x) = 3 x - 1$ and $g (x) = 2 x + 5$ .

$(3 x - 1) \frac{d}{d x} [2 x + 5] + (2 x + 5) \frac{d}{d x} [3 x - 1]$

Step 3.2

Differentiate.

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Step 3.2.1

By the Sum Rule, the derivative of $2 x + 5$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [5]$ .

$(3 x - 1) (\frac{d}{d x} [2 x] + \frac{d}{d x} [5]) + (2 x + 5) \frac{d}{d x} [3 x - 1]$

Step 3.2.2

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

$(3 x - 1) (2 \frac{d}{d x} [x] + \frac{d}{d x} [5]) + (2 x + 5) \frac{d}{d x} [3 x - 1]$

Step 3.2.3

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

$(3 x - 1) (2 \cdot 1 + \frac{d}{d x} [5]) + (2 x + 5) \frac{d}{d x} [3 x - 1]$

Step 3.2.4

Multiply $2$ by $1$ .

$(3 x - 1) (2 + \frac{d}{d x} [5]) + (2 x + 5) \frac{d}{d x} [3 x - 1]$

Step 3.2.5

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

$(3 x - 1) (2 + 0) + (2 x + 5) \frac{d}{d x} [3 x - 1]$

Step 3.2.6

Simplify the expression.

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Step 3.2.6.1

Add $2$ and $0$ .

$(3 x - 1) \cdot 2 + (2 x + 5) \frac{d}{d x} [3 x - 1]$

Step 3.2.6.2

Move $2$ to the left of $3 x - 1$ .

$2 \cdot (3 x - 1) + (2 x + 5) \frac{d}{d x} [3 x - 1]$

Step 3.2.7

By the Sum Rule, the derivative of $3 x - 1$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [- 1]$ .

$2 (3 x - 1) + (2 x + 5) (\frac{d}{d x} [3 x] + \frac{d}{d x} [- 1])$

Step 3.2.8

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$2 (3 x - 1) + (2 x + 5) (3 \frac{d}{d x} [x] + \frac{d}{d x} [- 1])$

Step 3.2.9

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

$2 (3 x - 1) + (2 x + 5) (3 \cdot 1 + \frac{d}{d x} [- 1])$

Step 3.2.10

Multiply $3$ by $1$ .

$2 (3 x - 1) + (2 x + 5) (3 + \frac{d}{d x} [- 1])$

Step 3.2.11

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$2 (3 x - 1) + (2 x + 5) (3 + 0)$

Step 3.2.12

Simplify the expression.

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Step 3.2.12.1

Add $3$ and $0$ .

$2 (3 x - 1) + (2 x + 5) \cdot 3$

Step 3.2.12.2

Move $3$ to the left of $2 x + 5$ .

$2 (3 x - 1) + 3 \cdot (2 x + 5)$

Step 3.3

Simplify.

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Step 3.3.1

Apply the distributive property.

$2 (3 x) + 2 \cdot - 1 + 3 (2 x + 5)$

Step 3.3.2

Apply the distributive property.

$2 (3 x) + 2 \cdot - 1 + 3 (2 x) + 3 \cdot 5$

Step 3.3.3

Combine terms.

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Step 3.3.3.1

Multiply $3$ by $2$ .

$6 x + 2 \cdot - 1 + 3 (2 x) + 3 \cdot 5$

Step 3.3.3.2

Multiply $2$ by $- 1$ .

$6 x - 2 + 3 (2 x) + 3 \cdot 5$

Step 3.3.3.3

Multiply $2$ by $3$ .

$6 x - 2 + 6 x + 3 \cdot 5$

Step 3.3.3.4

Multiply $3$ by $5$ .

$6 x - 2 + 6 x + 15$

Step 3.3.3.5

Add $6 x$ and $6 x$ .

$12 x - 2 + 15$

Step 3.3.3.6

Add $- 2$ and $15$ .

$12 x + 13$

Step 4

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

$y' = 12 x + 13$

Step 5

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

$\frac{d y}{d x} = 12 x + 13$