Calculus Examples

$y = 5 x^{2} + 5 x - 2$

Step 1

Find the first derivative.

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Step 1.1

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

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

Step 1.2

Evaluate $\frac{d}{d x} [5 x^{2}]$ .

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Step 1.2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{2}$ with respect to $x$ is $5 \frac{d}{d x} [x^{2}]$ .

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

Step 1.2.2

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

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

Step 1.2.3

Multiply $2$ by $5$ .

$10 x + \frac{d}{d x} [5 x] + \frac{d}{d x} [- 2]$

Step 1.3

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

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Step 1.3.1

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

$10 x + 5 \frac{d}{d x} [x] + \frac{d}{d x} [- 2]$

Step 1.3.2

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

$10 x + 5 \cdot 1 + \frac{d}{d x} [- 2]$

Step 1.3.3

Multiply $5$ by $1$ .

$10 x + 5 + \frac{d}{d x} [- 2]$

Step 1.4

Differentiate using the Constant Rule.

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Step 1.4.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$10 x + 5 + 0$

Step 1.4.2

Add $10 x + 5$ and $0$ .

$f' (x) = 10 x + 5$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $10 x + 5$ with respect to $x$ is $\frac{d}{d x} [10 x] + \frac{d}{d x} [5]$ .

$\frac{d}{d x} [10 x] + \frac{d}{d x} [5]$

Step 2.2

Evaluate $\frac{d}{d x} [10 x]$ .

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Step 2.2.1

Since $10$ is constant with respect to $x$ , the derivative of $10 x$ with respect to $x$ is $10 \frac{d}{d x} [x]$ .

$10 \frac{d}{d x} [x] + \frac{d}{d x} [5]$

Step 2.2.2

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

$10 \cdot 1 + \frac{d}{d x} [5]$

Step 2.2.3

Multiply $10$ by $1$ .

$10 + \frac{d}{d x} [5]$

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 $0$ .

$10 + 0$

Step 2.3.2

Add $10$ and $0$ .

$f'' (x) = 10$

Step 3

Since $10$ is constant with respect to $x$ , the derivative of $10$ with respect to $x$ is $0$ .

$f''' (x) = 0$

Step 4

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

$f^{4} (x) = 0$