Calculus Examples

$y = x - x^{3}$ , $(1, 0)$

Step 1

Differentiate.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

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

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Step 2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{3}$ with respect to $x$ is $- \frac{d}{d x} [x^{3}]$ .

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

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$1 - (3 x^{2})$

Step 2.3

Multiply $3$ by $- 1$ .

$1 - 3 x^{2}$

Step 3

Reorder terms.

$- 3 x^{2} + 1$

Step 4

Evaluate the derivative at $x = 1$ .

$- 3 {(1)}^{2} + 1$

Step 5

Simplify.

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Step 5.1

Simplify each term.

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Step 5.1.1

One to any power is one.

$- 3 \cdot 1 + 1$

Step 5.1.2

Multiply $- 3$ by $1$ .

$- 3 + 1$

Step 5.2

Add $- 3$ and $1$ .

$- 2$