Calculus Examples

$2 x - 5 x^{6}$

Step 1

Find the first derivative.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

Multiply $2$ by $1$ .

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

Step 1.3

Evaluate $\frac{d}{d x} [- 5 x^{6}]$ .

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Step 1.3.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x^{6}$ with respect to $x$ is $- 5 \frac{d}{d x} [x^{6}]$ .

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

Step 1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$2 - 5 (6 x^{5})$

Step 1.3.3

Multiply $6$ by $- 5$ .

$2 - 30 x^{5}$

Step 1.4

Reorder terms.

$f' (x) = - 30 x^{5} + 2$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $- 30 x^{5} + 2$ with respect to $x$ is $\frac{d}{d x} [- 30 x^{5}] + \frac{d}{d x} [2]$ .

$\frac{d}{d x} [- 30 x^{5}] + \frac{d}{d x} [2]$

Step 2.2

Evaluate $\frac{d}{d x} [- 30 x^{5}]$ .

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Step 2.2.1

Since $- 30$ is constant with respect to $x$ , the derivative of $- 30 x^{5}$ with respect to $x$ is $- 30 \frac{d}{d x} [x^{5}]$ .

$- 30 \frac{d}{d x} [x^{5}] + \frac{d}{d x} [2]$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$- 30 (5 x^{4}) + \frac{d}{d x} [2]$

Step 2.2.3

Multiply $5$ by $- 30$ .

$- 150 x^{4} + \frac{d}{d x} [2]$

Step 2.3

Differentiate using the Constant Rule.

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Step 2.3.1

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

$- 150 x^{4} + 0$

Step 2.3.2

Add $- 150 x^{4}$ and $0$ .

$f'' (x) = - 150 x^{4}$