Calculus Examples

$y = 5 x - 3$

Step 1

Differentiate both sides of the equation.

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

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

By the Sum Rule, the derivative of $5 x - 3$ with respect to $x$ is $\frac{d}{d x} [5 x] + \frac{d}{d x} [- 3]$ .

$\frac{d}{d x} [5 x] + \frac{d}{d x} [- 3]$

Step 3.2

Evaluate $\frac{d}{d x} [5 x]$ .

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Step 3.2.1

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

$5 \frac{d}{d x} [x] + \frac{d}{d x} [- 3]$

Step 3.2.2

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

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

Step 3.2.3

Multiply $5$ by $1$ .

$5 + \frac{d}{d x} [- 3]$

Step 3.3

Differentiate using the Constant Rule.

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Step 3.3.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$5 + 0$

Step 3.3.2

Add $5$ and $0$ .

$5$

Step 4

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

$y' = 5$

Step 5

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

$\frac{d y}{d x} = 5$