Calculus Examples

$2 (- 2 y + 2) + y + 2$

Step 1

This derivative could not be completed using the chain rule. Mathway will use another method.

Step 2

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

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

Step 3

Evaluate $\frac{d}{d y} [2 (- 2 y + 2)]$ .

Tap for more steps...

Step 3.1

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

$2 \frac{d}{d y} [- 2 y + 2] + \frac{d}{d y} [y] + \frac{d}{d y} [2]$

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

$2 (- 2 \cdot 1 + \frac{d}{d y} [2]) + \frac{d}{d y} [y] + \frac{d}{d y} [2]$

Step 3.5

Since $2$ is constant with respect to $y$ , the derivative of $2$ with respect to $y$ is $0$ .

$2 (- 2 \cdot 1 + 0) + \frac{d}{d y} [y] + \frac{d}{d y} [2]$

Step 3.6

Multiply $- 2$ by $1$ .

$2 (- 2 + 0) + \frac{d}{d y} [y] + \frac{d}{d y} [2]$

Step 3.7

Add $- 2$ and $0$ .

$2 \cdot - 2 + \frac{d}{d y} [y] + \frac{d}{d y} [2]$

Step 3.8

Multiply $2$ by $- 2$ .

$- 4 + \frac{d}{d y} [y] + \frac{d}{d y} [2]$

Step 4

Differentiate.

Tap for more steps...

Step 4.1

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

$- 4 + 1 + \frac{d}{d y} [2]$

Step 4.2

Since $2$ is constant with respect to $y$ , the derivative of $2$ with respect to $y$ is $0$ .

$- 4 + 1 + 0$

Step 5

Combine terms.

Tap for more steps...

Step 5.1

Add $- 4$ and $1$ .

$- 3 + 0$

Step 5.2

Add $- 3$ and $0$ .

$- 3$