Calculus Examples

$f (x) = 5 + 75 x - 2 x^{2} + 8 y - 0.5 x y + 4 z^{2}$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $5 + 75 x - 2 x^{2} + 8 y - 0.5 x y + 4 z^{2}$ with respect to $x$ is $\frac{d}{d x} [5] + \frac{d}{d x} [75 x] + \frac{d}{d x} [- 2 x^{2}] + \frac{d}{d x} [8 y] + \frac{d}{d x} [- 0.5 x y] + \frac{d}{d x} [4 z^{2}]$ .

$\frac{d}{d x} [5] + \frac{d}{d x} [75 x] + \frac{d}{d x} [- 2 x^{2}] + \frac{d}{d x} [8 y] + \frac{d}{d x} [- 0.5 x y] + \frac{d}{d x} [4 z^{2}]$

Step 1.2

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

$0 + \frac{d}{d x} [75 x] + \frac{d}{d x} [- 2 x^{2}] + \frac{d}{d x} [8 y] + \frac{d}{d x} [- 0.5 x y] + \frac{d}{d x} [4 z^{2}]$

Step 2

Evaluate $\frac{d}{d x} [75 x]$ .

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Step 2.1

Since $75$ is constant with respect to $x$ , the derivative of $75 x$ with respect to $x$ is $75 \frac{d}{d x} [x]$ .

$0 + 75 \frac{d}{d x} [x] + \frac{d}{d x} [- 2 x^{2}] + \frac{d}{d x} [8 y] + \frac{d}{d x} [- 0.5 x y] + \frac{d}{d x} [4 z^{2}]$

Step 2.2

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

$0 + 75 \cdot 1 + \frac{d}{d x} [- 2 x^{2}] + \frac{d}{d x} [8 y] + \frac{d}{d x} [- 0.5 x y] + \frac{d}{d x} [4 z^{2}]$

Step 2.3

Multiply $75$ by $1$ .

$0 + 75 + \frac{d}{d x} [- 2 x^{2}] + \frac{d}{d x} [8 y] + \frac{d}{d x} [- 0.5 x y] + \frac{d}{d x} [4 z^{2}]$

Step 3

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

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Step 3.1

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

$0 + 75 - 2 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [8 y] + \frac{d}{d x} [- 0.5 x y] + \frac{d}{d x} [4 z^{2}]$

Step 3.2

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

$0 + 75 - 2 (2 x) + \frac{d}{d x} [8 y] + \frac{d}{d x} [- 0.5 x y] + \frac{d}{d x} [4 z^{2}]$

Step 3.3

Multiply $2$ by $- 2$ .

$0 + 75 - 4 x + \frac{d}{d x} [8 y] + \frac{d}{d x} [- 0.5 x y] + \frac{d}{d x} [4 z^{2}]$

Step 4

Since $8 y$ is constant with respect to $x$ , the derivative of $8 y$ with respect to $x$ is $0$ .

$0 + 75 - 4 x + 0 + \frac{d}{d x} [- 0.5 x y] + \frac{d}{d x} [4 z^{2}]$

Step 5

Evaluate $\frac{d}{d x} [- 0.5 x y]$ .

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Step 5.1

Since $- 0.5 y$ is constant with respect to $x$ , the derivative of $- 0.5 x y$ with respect to $x$ is $- 0.5 y \frac{d}{d x} [x]$ .

$0 + 75 - 4 x + 0 - 0.5 y \frac{d}{d x} [x] + \frac{d}{d x} [4 z^{2}]$

Step 5.2

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

$0 + 75 - 4 x + 0 - 0.5 y \cdot 1 + \frac{d}{d x} [4 z^{2}]$

Step 5.3

Multiply $- 0.5$ by $1$ .

$0 + 75 - 4 x + 0 - 0.5 y + \frac{d}{d x} [4 z^{2}]$

Step 6

Since $4 z^{2}$ is constant with respect to $x$ , the derivative of $4 z^{2}$ with respect to $x$ is $0$ .

$0 + 75 - 4 x + 0 - 0.5 y + 0$

Step 7

Simplify.

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Step 7.1

Combine terms.

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Step 7.1.1

Add $0$ and $75$ .

$75 - 4 x + 0 - 0.5 y + 0$

Step 7.1.2

Add $75 - 4 x$ and $0$ .

$75 - 4 x - 0.5 y + 0$

Step 7.1.3

Add $75 - 4 x - 0.5 y$ and $0$ .

$75 - 4 x - 0.5 y$

Step 7.2

Reorder terms.

$- 4 x - 0.5 y + 75$