Calculus Examples

$c (y) = 240000 + 6000 x + 4000 y$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $240000 + 6000 x + 4000 y$ with respect to $x$ is $\frac{d}{d x} [240000] + \frac{d}{d x} [6000 x] + \frac{d}{d x} [4000 y]$ .

$\frac{d}{d x} [240000] + \frac{d}{d x} [6000 x] + \frac{d}{d x} [4000 y]$

Step 1.2

Since $240000$ is constant with respect to $x$ , the derivative of $240000$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [6000 x] + \frac{d}{d x} [4000 y]$

Step 2

Evaluate $\frac{d}{d x} [6000 x]$ .

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Step 2.1

Since $6000$ is constant with respect to $x$ , the derivative of $6000 x$ with respect to $x$ is $6000 \frac{d}{d x} [x]$ .

$0 + 6000 \frac{d}{d x} [x] + \frac{d}{d x} [4000 y]$

Step 2.2

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

$0 + 6000 \cdot 1 + \frac{d}{d x} [4000 y]$

Step 2.3

Multiply $6000$ by $1$ .

$0 + 6000 + \frac{d}{d x} [4000 y]$

Step 3

Since $4000 y$ is constant with respect to $x$ , the derivative of $4000 y$ with respect to $x$ is $0$ .

$0 + 6000 + 0$

Step 4

Combine terms.

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Step 4.1

Add $0$ and $6000$ .

$6000 + 0$

Step 4.2

Add $6000$ and $0$ .

$6000$