Calculus Examples

$y = x - x^{2}$

Step 1

Write $y = x - x^{2}$ as a function.

$f (x) = x - x^{2}$

Step 2

Find the second derivative.

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Step 2.1

Find the first derivative.

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Step 2.1.1

Differentiate.

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Step 2.1.1.1

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

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

Step 2.1.1.2

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

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

Step 2.1.2

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

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Step 2.1.2.1

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

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

Step 2.1.2.2

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

$1 - (2 x)$

Step 2.1.2.3

Multiply $2$ by $- 1$ .

$1 - 2 x$

Step 2.1.3

Reorder terms.

$f' (x) = - 2 x + 1$

Step 2.2

Find the second derivative.

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Step 2.2.1

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

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

Step 2.2.2

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

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Step 2.2.2.1

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

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

Step 2.2.2.2

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

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

Step 2.2.2.3

Multiply $- 2$ by $1$ .

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

Step 2.2.3

Differentiate using the Constant Rule.

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Step 2.2.3.1

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$- 2 + 0$

Step 2.2.3.2

Add $- 2$ and $0$ .

$f'' (x) = - 2$

Step 2.3

The second derivative of $f (x)$ with respect to $x$ is $- 2$ .

$- 2$

Step 3

Set the second derivative equal to $0$ then solve the equation $- 2 = 0$ .

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Step 3.1

Set the second derivative equal to $0$ .

$- 2 = 0$

Step 3.2

Since $- 2 \neq 0$ , there are no solutions.

No solution

Step 4

No values found that can make the second derivative equal to $0$ .

No Inflection Points