Calculus Examples

$x^{2} - 5 x + 6$

Step 1

Find the first derivative.

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Step 1.1

Differentiate.

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Step 1.1.1

By the Sum Rule, the derivative of $x^{2} - 5 x + 6$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [6]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [6]$

Step 1.1.2

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

$2 x + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [6]$

Step 1.2

Evaluate $\frac{d}{d x} [- 5 x]$ .

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Step 1.2.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x$ with respect to $x$ is $- 5 \frac{d}{d x} [x]$ .

$2 x - 5 \frac{d}{d x} [x] + \frac{d}{d x} [6]$

Step 1.2.2

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

$2 x - 5 \cdot 1 + \frac{d}{d x} [6]$

Step 1.2.3

Multiply $- 5$ by $1$ .

$2 x - 5 + \frac{d}{d x} [6]$

Step 1.3

Differentiate using the Constant Rule.

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Step 1.3.1

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$2 x - 5 + 0$

Step 1.3.2

Add $2 x - 5$ and $0$ .

$f' (x) = 2 x - 5$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $2 x - 5$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- 5]$ .

$\frac{d}{d x} [2 x] + \frac{d}{d x} [- 5]$

Step 2.2

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

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Step 2.2.1

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

$2 \frac{d}{d x} [x] + \frac{d}{d x} [- 5]$

Step 2.2.2

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

$2 \cdot 1 + \frac{d}{d x} [- 5]$

Step 2.2.3

Multiply $2$ by $1$ .

$2 + \frac{d}{d x} [- 5]$

Step 2.3

Differentiate using the Constant Rule.

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Step 2.3.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$2 + 0$

Step 2.3.2

Add $2$ and $0$ .

$f'' (x) = 2$

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