Calculus Examples

$x^{2} + x y + y^{2} + 2 x - y$

Step 1

Differentiate.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

$2 x + y \cdot 1 + \frac{d}{d x} [y^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [- y]$

Step 2.3

Multiply $y$ by $1$ .

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Multiply $2$ by $1$ .

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

Step 5

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

$2 x + y + 0 + 2 + 0$

Step 6

Combine terms.

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Step 6.1

Add $2 x + y$ and $0$ .

$2 x + y + 2 + 0$

Step 6.2

Add $2 x + y + 2$ and $0$ .

$2 x + y + 2$