Calculus Examples

$z (y) = x^{2} + y^{2} - L (2 x + y + 1)$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $x^{2} + y^{2} - L (2 x + y + 1)$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [y^{2}] + \frac{d}{d x} [- L (2 x + y + 1)]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [y^{2}] + \frac{d}{d x} [- L (2 x + y + 1)]$

Step 1.2

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

$2 x + \frac{d}{d x} [y^{2}] + \frac{d}{d x} [- L (2 x + y + 1)]$

Step 1.3

Since $y^{2}$ is constant with respect to $x$ , the derivative of $y^{2}$ with respect to $x$ is $0$ .

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

Step 2

Evaluate $\frac{d}{d x} [- L (2 x + y + 1)]$ .

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Step 2.1

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

$2 x + 0 - L \frac{d}{d x} [2 x + y + 1]$

Step 2.2

By the Sum Rule, the derivative of $2 x + y + 1$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [y] + \frac{d}{d x} [1]$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$2 x + 0 - L (2 \cdot 1 + 0 + 0)$

Step 2.7

Multiply $2$ by $1$ .

$2 x + 0 - L (2 + 0 + 0)$

Step 2.8

Add $2$ and $0$ .

$2 x + 0 - L (2 + 0)$

Step 2.9

Add $2$ and $0$ .

$2 x + 0 - L \cdot 2$

Step 2.10

Multiply $2$ by $- 1$ .

$2 x + 0 - 2 L$

Step 3

Simplify.

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Step 3.1

Add $2 x$ and $0$ .

$2 x - 2 L$

Step 3.2

Reorder terms.

$- 2 L + 2 x$