Calculus Examples

$- 5 x^{2} + 75 x - 30$

Step 1

By the Sum Rule, the derivative of $- 5 x^{2} + 75 x - 30$ with respect to $x$ is $\frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [75 x] + \frac{d}{d x} [- 30]$ .

$\frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [75 x] + \frac{d}{d x} [- 30]$

Step 2

Evaluate $\frac{d}{d x} [- 5 x^{2}]$ .

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Step 2.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x^{2}$ with respect to $x$ is $- 5 \frac{d}{d x} [x^{2}]$ .

$- 5 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [75 x] + \frac{d}{d x} [- 30]$

Step 2.2

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

$- 5 (2 x) + \frac{d}{d x} [75 x] + \frac{d}{d x} [- 30]$

Step 2.3

Multiply $2$ by $- 5$ .

$- 10 x + \frac{d}{d x} [75 x] + \frac{d}{d x} [- 30]$

Step 3

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

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Step 3.1

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

$- 10 x + 75 \frac{d}{d x} [x] + \frac{d}{d x} [- 30]$

Step 3.2

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

$- 10 x + 75 \cdot 1 + \frac{d}{d x} [- 30]$

Step 3.3

Multiply $75$ by $1$ .

$- 10 x + 75 + \frac{d}{d x} [- 30]$

Step 4

Differentiate using the Constant Rule.

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Step 4.1

Since $- 30$ is constant with respect to $x$ , the derivative of $- 30$ with respect to $x$ is $0$ .

$- 10 x + 75 + 0$

Step 4.2

Add $- 10 x + 75$ and $0$ .

$- 10 x + 75$