Calculus Examples

$2 x^{10} - x^{6} + 5 x^{2}$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $2 x^{10} - x^{6} + 5 x^{2}$ with respect to $x$ is $\frac{d}{d x} [2 x^{10}] + \frac{d}{d x} [- x^{6}] + \frac{d}{d x} [5 x^{2}]$ .

$\frac{d}{d x} [2 x^{10}] + \frac{d}{d x} [- x^{6}] + \frac{d}{d x} [5 x^{2}]$

Step 1.2

Evaluate $\frac{d}{d x} [2 x^{10}]$ .

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Step 1.2.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{10}$ with respect to $x$ is $2 \frac{d}{d x} [x^{10}]$ .

$2 \frac{d}{d x} [x^{10}] + \frac{d}{d x} [- x^{6}] + \frac{d}{d x} [5 x^{2}]$

Step 1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 10$ .

$2 (10 x^{9}) + \frac{d}{d x} [- x^{6}] + \frac{d}{d x} [5 x^{2}]$

Step 1.2.3

Multiply $10$ by $2$ .

$20 x^{9} + \frac{d}{d x} [- x^{6}] + \frac{d}{d x} [5 x^{2}]$

Step 1.3

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

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Step 1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{6}$ with respect to $x$ is $- \frac{d}{d x} [x^{6}]$ .

$20 x^{9} - \frac{d}{d x} [x^{6}] + \frac{d}{d x} [5 x^{2}]$

Step 1.3.2

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

$20 x^{9} - (6 x^{5}) + \frac{d}{d x} [5 x^{2}]$

Step 1.3.3

Multiply $6$ by $- 1$ .

$20 x^{9} - 6 x^{5} + \frac{d}{d x} [5 x^{2}]$

Step 1.4

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

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Step 1.4.1

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

$20 x^{9} - 6 x^{5} + 5 \frac{d}{d x} [x^{2}]$

Step 1.4.2

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

$20 x^{9} - 6 x^{5} + 5 (2 x)$

Step 1.4.3

Multiply $2$ by $5$ .

$f' (x) = 20 x^{9} - 6 x^{5} + 10 x$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $20 x^{9} - 6 x^{5} + 10 x$ with respect to $x$ is $\frac{d}{d x} [20 x^{9}] + \frac{d}{d x} [- 6 x^{5}] + \frac{d}{d x} [10 x]$ .

$\frac{d}{d x} [20 x^{9}] + \frac{d}{d x} [- 6 x^{5}] + \frac{d}{d x} [10 x]$

Step 2.2

Evaluate $\frac{d}{d x} [20 x^{9}]$ .

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Step 2.2.1

Since $20$ is constant with respect to $x$ , the derivative of $20 x^{9}$ with respect to $x$ is $20 \frac{d}{d x} [x^{9}]$ .

$20 \frac{d}{d x} [x^{9}] + \frac{d}{d x} [- 6 x^{5}] + \frac{d}{d x} [10 x]$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 9$ .

$20 (9 x^{8}) + \frac{d}{d x} [- 6 x^{5}] + \frac{d}{d x} [10 x]$

Step 2.2.3

Multiply $9$ by $20$ .

$180 x^{8} + \frac{d}{d x} [- 6 x^{5}] + \frac{d}{d x} [10 x]$

Step 2.3

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

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Step 2.3.1

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

$180 x^{8} - 6 \frac{d}{d x} [x^{5}] + \frac{d}{d x} [10 x]$

Step 2.3.2

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

$180 x^{8} - 6 (5 x^{4}) + \frac{d}{d x} [10 x]$

Step 2.3.3

Multiply $5$ by $- 6$ .

$180 x^{8} - 30 x^{4} + \frac{d}{d x} [10 x]$

Step 2.4

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

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Step 2.4.1

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

$180 x^{8} - 30 x^{4} + 10 \frac{d}{d x} [x]$

Step 2.4.2

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

$180 x^{8} - 30 x^{4} + 10 \cdot 1$

Step 2.4.3

Multiply $10$ by $1$ .

$f'' (x) = 180 x^{8} - 30 x^{4} + 10$