Calculus Examples

$p (x) = \frac{- 1}{2} x^{2} + 3 x - 100$

Step 1

Differentiate using the Sum Rule.

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Step 1.1

Move the negative in front of the fraction.

$\frac{d}{d x} [- \frac{1}{2} x^{2} + 3 x - 100]$

Step 1.2

By the Sum Rule, the derivative of $- \frac{1}{2} x^{2} + 3 x - 100$ with respect to $x$ is $\frac{d}{d x} [- \frac{1}{2} x^{2}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 100]$ .

$\frac{d}{d x} [- \frac{1}{2} x^{2}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 100]$

Step 2

Evaluate $\frac{d}{d x} [- \frac{1}{2} x^{2}]$ .

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Step 2.1

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

$- \frac{1}{2} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 100]$

Step 2.2

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

$- \frac{1}{2} (2 x) + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 100]$

Step 2.3

Multiply $2$ by $- 1$ .

$- 2 (\frac{1}{2}) x + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 100]$

Step 2.4

Combine $- 2$ and $\frac{1}{2}$ .

$\frac{- 2}{2} x + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 100]$

Step 2.5

Combine $\frac{- 2}{2}$ and $x$ .

$\frac{- 2 x}{2} + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 100]$

Step 2.6

Cancel the common factor of $- 2$ and $2$ .

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Step 2.6.1

Factor $2$ out of $- 2 x$ .

$\frac{2 (- x)}{2} + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 100]$

Step 2.6.2

Cancel the common factors.

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Step 2.6.2.1

Factor $2$ out of $2$ .

$\frac{2 (- x)}{2 (1)} + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 100]$

Step 2.6.2.2

Cancel the common factor.

$\frac{2 (- x)}{2 \cdot 1} + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 100]$

Step 2.6.2.3

Rewrite the expression.

$\frac{- x}{1} + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 100]$

Step 2.6.2.4

Divide $- x$ by $1$ .

$- x + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 100]$

Step 3

Evaluate $\frac{d}{d x} [3 x]$ .

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Step 3.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$- x + 3 \frac{d}{d x} [x] + \frac{d}{d x} [- 100]$

Step 3.2

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

$- x + 3 \cdot 1 + \frac{d}{d x} [- 100]$

Step 3.3

Multiply $3$ by $1$ .

$- x + 3 + \frac{d}{d x} [- 100]$

Step 4

Differentiate using the Constant Rule.

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Step 4.1

Since $- 100$ is constant with respect to $x$ , the derivative of $- 100$ with respect to $x$ is $0$ .

$- x + 3 + 0$

Step 4.2

Add $- x + 3$ and $0$ .

$- x + 3$