Calculus Examples

$y = 2 (x^{2} - 3 x)$

Step 1

Differentiate both sides of the equation.

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

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

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

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

Step 3.2

By the Sum Rule, the derivative of $x^{2} - 3 x$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3 x]$ .

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

Multiply $- 3$ by $1$ .

$2 (2 x - 3)$

Step 3.7

Simplify.

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Step 3.7.1

Apply the distributive property.

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

Step 3.7.2

Combine terms.

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Step 3.7.2.1

Multiply $2$ by $2$ .

$4 x + 2 \cdot - 3$

Step 3.7.2.2

Multiply $2$ by $- 3$ .

$4 x - 6$

Step 4

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

$y' = 4 x - 6$

Step 5

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

$\frac{d y}{d x} = 4 x - 6$