Calculus Examples

$y = 4 x - x^{2}$ , $(1, 3)$

Step 1

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

$\frac{d}{d x} [4 x] + \frac{d}{d x} [- x^{2}]$

Step 2

Evaluate $\frac{d}{d x} [4 x]$ .

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Step 2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$4 \frac{d}{d x} [x] + \frac{d}{d x} [- x^{2}]$

Step 2.2

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

$4 \cdot 1 + \frac{d}{d x} [- x^{2}]$

Step 2.3

Multiply $4$ by $1$ .

$4 + \frac{d}{d x} [- x^{2}]$

Step 3

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

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Step 3.1

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

$4 - \frac{d}{d x} [x^{2}]$

Step 3.2

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

$4 - (2 x)$

Step 3.3

Multiply $2$ by $- 1$ .

$4 - 2 x$

Step 4

Reorder terms.

$- 2 x + 4$

Step 5

Evaluate the derivative at $x = 1$ .

$- 2 \cdot 1 + 4$

Step 6

Simplify.

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Step 6.1

Multiply $- 2$ by $1$ .

$- 2 + 4$

Step 6.2

Add $- 2$ and $4$ .

$2$