Calculus Examples

$5 x y - x^{2} - y^{2} - 5$

Step 1

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

$\frac{d}{d x} [5 x y] + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [- y^{2}] + \frac{d}{d x} [- 5]$

Step 2

Evaluate $\frac{d}{d x} [5 x y]$ .

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Step 2.1

Since $5 y$ is constant with respect to $x$ , the derivative of $5 x y$ with respect to $x$ is $5 y \frac{d}{d x} [x]$ .

$5 y \frac{d}{d x} [x] + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [- y^{2}] + \frac{d}{d x} [- 5]$

Step 2.2

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

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

Step 2.3

Multiply $5$ by $1$ .

$5 y + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [- y^{2}] + \frac{d}{d x} [- 5]$

Step 3

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

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Step 3.1

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

$5 y - \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- y^{2}] + \frac{d}{d x} [- 5]$

Step 3.2

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

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

Step 3.3

Multiply $2$ by $- 1$ .

$5 y - 2 x + \frac{d}{d x} [- y^{2}] + \frac{d}{d x} [- 5]$

Step 4

Differentiate using the Constant Rule.

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Step 4.1

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

$5 y - 2 x + 0 + \frac{d}{d x} [- 5]$

Step 4.2

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$5 y - 2 x + 0 + 0$

Step 5

Simplify.

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Step 5.1

Combine terms.

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Step 5.1.1

Add $5 y - 2 x$ and $0$ .

$5 y - 2 x + 0$

Step 5.1.2

Add $5 y - 2 x$ and $0$ .

$5 y - 2 x$

Step 5.2

Reorder terms.

$- 2 x + 5 y$