Calculus Examples

$- x^{2} + 4 x y - 5 y^{2} - 6 x + 22 y + 9$

Step 1

By the Sum Rule, the derivative of $- x^{2} + 4 x y - 5 y^{2} - 6 x + 22 y + 9$ with respect to $x$ is $\frac{d}{d x} [- x^{2}] + \frac{d}{d x} [4 x y] + \frac{d}{d x} [- 5 y^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [22 y] + \frac{d}{d x} [9]$ .

$\frac{d}{d x} [- x^{2}] + \frac{d}{d x} [4 x y] + \frac{d}{d x} [- 5 y^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [22 y] + \frac{d}{d x} [9]$

Step 2

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

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Step 2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$- \frac{d}{d x} [x^{2}] + \frac{d}{d x} [4 x y] + \frac{d}{d x} [- 5 y^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [22 y] + \frac{d}{d x} [9]$

Step 2.2

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

$- (2 x) + \frac{d}{d x} [4 x y] + \frac{d}{d x} [- 5 y^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [22 y] + \frac{d}{d x} [9]$

Step 2.3

Multiply $2$ by $- 1$ .

$- 2 x + \frac{d}{d x} [4 x y] + \frac{d}{d x} [- 5 y^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [22 y] + \frac{d}{d x} [9]$

Step 3

Evaluate $\frac{d}{d x} [4 x y]$ .

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Step 3.1

Since $4 y$ is constant with respect to $x$ , the derivative of $4 x y$ with respect to $x$ is $4 y \frac{d}{d x} [x]$ .

$- 2 x + 4 y \frac{d}{d x} [x] + \frac{d}{d x} [- 5 y^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [22 y] + \frac{d}{d x} [9]$

Step 3.2

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

$- 2 x + 4 y \cdot 1 + \frac{d}{d x} [- 5 y^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [22 y] + \frac{d}{d x} [9]$

Step 3.3

Multiply $4$ by $1$ .

$- 2 x + 4 y + \frac{d}{d x} [- 5 y^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [22 y] + \frac{d}{d x} [9]$

Step 4

Since $- 5 y^{2}$ is constant with respect to $x$ , the derivative of $- 5 y^{2}$ with respect to $x$ is $0$ .

$- 2 x + 4 y + 0 + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [22 y] + \frac{d}{d x} [9]$

Step 5

Evaluate $\frac{d}{d x} [- 6 x]$ .

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Step 5.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x$ with respect to $x$ is $- 6 \frac{d}{d x} [x]$ .

$- 2 x + 4 y + 0 - 6 \frac{d}{d x} [x] + \frac{d}{d x} [22 y] + \frac{d}{d x} [9]$

Step 5.2

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

$- 2 x + 4 y + 0 - 6 \cdot 1 + \frac{d}{d x} [22 y] + \frac{d}{d x} [9]$

Step 5.3

Multiply $- 6$ by $1$ .

$- 2 x + 4 y + 0 - 6 + \frac{d}{d x} [22 y] + \frac{d}{d x} [9]$

Step 6

Differentiate using the Constant Rule.

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Step 6.1

Since $22 y$ is constant with respect to $x$ , the derivative of $22 y$ with respect to $x$ is $0$ .

$- 2 x + 4 y + 0 - 6 + 0 + \frac{d}{d x} [9]$

Step 6.2

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$- 2 x + 4 y + 0 - 6 + 0 + 0$

Step 7

Combine terms.

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Step 7.1

Add $- 2 x + 4 y$ and $0$ .

$- 2 x + 4 y - 6 + 0 + 0$

Step 7.2

Add $- 2 x + 4 y - 6$ and $0$ .

$- 2 x + 4 y - 6 + 0$

Step 7.3

Add $- 2 x + 4 y - 6$ and $0$ .

$- 2 x + 4 y - 6$