Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

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 $3 x^{2} - 5 x$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 5 x]$ .

$\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 5 x]$

Step 3.2

Evaluate $\frac{d}{d x} [3 x^{2}]$ .

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Step 3.2.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

$3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 5 x]$

Step 3.2.2

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

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

Step 3.2.3

Multiply $2$ by $3$ .

$6 x + \frac{d}{d x} [- 5 x]$

Step 3.3

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

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Step 3.3.1

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

$6 x - 5 \frac{d}{d x} [x]$

Step 3.3.2

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

$6 x - 5 \cdot 1$

Step 3.3.3

Multiply $- 5$ by $1$ .

$6 x - 5$

Step 4

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

$y' = 6 x - 5$

Step 5

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

$\frac{d y}{d x} = 6 x - 5$