Calculus Examples

$f (x) = - 2 x^{2} - 3 x$ , $x = 1$

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Multiply $2$ by $- 2$ .

$- 4 x + \frac{d}{d x} [- 3 x]$

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

$- 4 x - 3 \frac{d}{d x} [x]$

Step 3.2

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

$- 4 x - 3 \cdot 1$

Step 3.3

Multiply $- 3$ by $1$ .

$- 4 x - 3$

Step 4

Evaluate the derivative at $x = 1$ .

$- 4 \cdot 1 - 3$

Step 5

Simplify.

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Step 5.1

Multiply $- 4$ by $1$ .

$- 4 - 3$

Step 5.2

Subtract $3$ from $- 4$ .

$- 7$