Calculus Examples

$y = \frac{2 x^{2} - 5 x}{3}$

Step 1

Differentiate both sides of the equation.

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

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 Constant Multiple Rule.

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Step 3.1.1

Factor $x$ out of $2 x^{2} - 5 x$ .

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Step 3.1.1.1

Factor $x$ out of $2 x^{2}$ .

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

Step 3.1.1.2

Factor $x$ out of $- 5 x$ .

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

Step 3.1.1.3

Factor $x$ out of $x (2 x) + x \cdot - 5$ .

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

Step 3.1.2

Since $\frac{1}{3}$ is constant with respect to $x$ , the derivative of $\frac{x (2 x - 5)}{3}$ with respect to $x$ is $\frac{1}{3} \frac{d}{d x} [x (2 x - 5)]$ .

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

Step 3.2

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 $g (x) = 2 x - 5$ .

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

Step 3.3

Differentiate.

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.3.4

Multiply $2$ by $1$ .

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

Step 3.3.5

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

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

Step 3.3.6

Simplify the expression.

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Step 3.3.6.1

Add $2$ and $0$ .

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

Step 3.3.6.2

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

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

Step 3.3.7

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

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

Step 3.3.8

Simplify by adding terms.

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Step 3.3.8.1

Multiply $2 x - 5$ by $1$ .

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

Step 3.3.8.2

Add $2 x$ and $2 x$ .

$\frac{1}{3} (4 x - 5)$

Step 3.4

Simplify.

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Step 3.4.1

Apply the distributive property.

$\frac{1}{3} (4 x) + \frac{1}{3} \cdot - 5$

Step 3.4.2

Combine terms.

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Step 3.4.2.1

Combine $4$ and $\frac{1}{3}$ .

$\frac{4}{3} x + \frac{1}{3} \cdot - 5$

Step 3.4.2.2

Combine $\frac{4}{3}$ and $x$ .

$\frac{4 x}{3} + \frac{1}{3} \cdot - 5$

Step 3.4.2.3

Combine $\frac{1}{3}$ and $- 5$ .

$\frac{4 x}{3} + \frac{- 5}{3}$

Step 3.4.2.4

Move the negative in front of the fraction.

$\frac{4 x}{3} - \frac{5}{3}$

Step 4

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

$y' = \frac{4 x}{3} - \frac{5}{3}$

Step 5

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

$\frac{d y}{d x} = \frac{4 x}{3} - \frac{5}{3}$