Calculus Examples

$3 x^{5} + 4 x^{4} - 18 x^{2} + 6 x$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $3 x^{5} + 4 x^{4} - 18 x^{2} + 6 x$ with respect to $x$ is $\frac{d}{d x} [3 x^{5}] + \frac{d}{d x} [4 x^{4}] + \frac{d}{d x} [- 18 x^{2}] + \frac{d}{d x} [6 x]$ .

$\frac{d}{d x} [3 x^{5}] + \frac{d}{d x} [4 x^{4}] + \frac{d}{d x} [- 18 x^{2}] + \frac{d}{d x} [6 x]$

Step 1.2

Evaluate $\frac{d}{d x} [3 x^{5}]$ .

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Step 1.2.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{5}$ with respect to $x$ is $3 \frac{d}{d x} [x^{5}]$ .

$3 \frac{d}{d x} [x^{5}] + \frac{d}{d x} [4 x^{4}] + \frac{d}{d x} [- 18 x^{2}] + \frac{d}{d x} [6 x]$

Step 1.2.2

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

$3 (5 x^{4}) + \frac{d}{d x} [4 x^{4}] + \frac{d}{d x} [- 18 x^{2}] + \frac{d}{d x} [6 x]$

Step 1.2.3

Multiply $5$ by $3$ .

$15 x^{4} + \frac{d}{d x} [4 x^{4}] + \frac{d}{d x} [- 18 x^{2}] + \frac{d}{d x} [6 x]$

Step 1.3

Evaluate $\frac{d}{d x} [4 x^{4}]$ .

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Step 1.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{4}$ with respect to $x$ is $4 \frac{d}{d x} [x^{4}]$ .

$15 x^{4} + 4 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 18 x^{2}] + \frac{d}{d x} [6 x]$

Step 1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$15 x^{4} + 4 (4 x^{3}) + \frac{d}{d x} [- 18 x^{2}] + \frac{d}{d x} [6 x]$

Step 1.3.3

Multiply $4$ by $4$ .

$15 x^{4} + 16 x^{3} + \frac{d}{d x} [- 18 x^{2}] + \frac{d}{d x} [6 x]$

Step 1.4

Evaluate $\frac{d}{d x} [- 18 x^{2}]$ .

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Step 1.4.1

Since $- 18$ is constant with respect to $x$ , the derivative of $- 18 x^{2}$ with respect to $x$ is $- 18 \frac{d}{d x} [x^{2}]$ .

$15 x^{4} + 16 x^{3} - 18 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [6 x]$

Step 1.4.2

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

$15 x^{4} + 16 x^{3} - 18 (2 x) + \frac{d}{d x} [6 x]$

Step 1.4.3

Multiply $2$ by $- 18$ .

$15 x^{4} + 16 x^{3} - 36 x + \frac{d}{d x} [6 x]$

Step 1.5

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

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Step 1.5.1

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

$15 x^{4} + 16 x^{3} - 36 x + 6 \frac{d}{d x} [x]$

Step 1.5.2

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

$15 x^{4} + 16 x^{3} - 36 x + 6 \cdot 1$

Step 1.5.3

Multiply $6$ by $1$ .

$f' (x) = 15 x^{4} + 16 x^{3} - 36 x + 6$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $15 x^{4} + 16 x^{3} - 36 x + 6$ with respect to $x$ is $\frac{d}{d x} [15 x^{4}] + \frac{d}{d x} [16 x^{3}] + \frac{d}{d x} [- 36 x] + \frac{d}{d x} [6]$ .

$\frac{d}{d x} [15 x^{4}] + \frac{d}{d x} [16 x^{3}] + \frac{d}{d x} [- 36 x] + \frac{d}{d x} [6]$

Step 2.2

Evaluate $\frac{d}{d x} [15 x^{4}]$ .

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Step 2.2.1

Since $15$ is constant with respect to $x$ , the derivative of $15 x^{4}$ with respect to $x$ is $15 \frac{d}{d x} [x^{4}]$ .

$15 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [16 x^{3}] + \frac{d}{d x} [- 36 x] + \frac{d}{d x} [6]$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$15 (4 x^{3}) + \frac{d}{d x} [16 x^{3}] + \frac{d}{d x} [- 36 x] + \frac{d}{d x} [6]$

Step 2.2.3

Multiply $4$ by $15$ .

$60 x^{3} + \frac{d}{d x} [16 x^{3}] + \frac{d}{d x} [- 36 x] + \frac{d}{d x} [6]$

Step 2.3

Evaluate $\frac{d}{d x} [16 x^{3}]$ .

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Step 2.3.1

Since $16$ is constant with respect to $x$ , the derivative of $16 x^{3}$ with respect to $x$ is $16 \frac{d}{d x} [x^{3}]$ .

$60 x^{3} + 16 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 36 x] + \frac{d}{d x} [6]$

Step 2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$60 x^{3} + 16 (3 x^{2}) + \frac{d}{d x} [- 36 x] + \frac{d}{d x} [6]$

Step 2.3.3

Multiply $3$ by $16$ .

$60 x^{3} + 48 x^{2} + \frac{d}{d x} [- 36 x] + \frac{d}{d x} [6]$

Step 2.4

Evaluate $\frac{d}{d x} [- 36 x]$ .

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Step 2.4.1

Since $- 36$ is constant with respect to $x$ , the derivative of $- 36 x$ with respect to $x$ is $- 36 \frac{d}{d x} [x]$ .

$60 x^{3} + 48 x^{2} - 36 \frac{d}{d x} [x] + \frac{d}{d x} [6]$

Step 2.4.2

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

$60 x^{3} + 48 x^{2} - 36 \cdot 1 + \frac{d}{d x} [6]$

Step 2.4.3

Multiply $- 36$ by $1$ .

$60 x^{3} + 48 x^{2} - 36 + \frac{d}{d x} [6]$

Step 2.5

Differentiate using the Constant Rule.

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Step 2.5.1

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

$60 x^{3} + 48 x^{2} - 36 + 0$

Step 2.5.2

Add $60 x^{3} + 48 x^{2} - 36$ and $0$ .

$f'' (x) = 60 x^{3} + 48 x^{2} - 36$

Step 3

Find the third derivative.

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Step 3.1

By the Sum Rule, the derivative of $60 x^{3} + 48 x^{2} - 36$ with respect to $x$ is $\frac{d}{d x} [60 x^{3}] + \frac{d}{d x} [48 x^{2}] + \frac{d}{d x} [- 36]$ .

$\frac{d}{d x} [60 x^{3}] + \frac{d}{d x} [48 x^{2}] + \frac{d}{d x} [- 36]$

Step 3.2

Evaluate $\frac{d}{d x} [60 x^{3}]$ .

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Step 3.2.1

Since $60$ is constant with respect to $x$ , the derivative of $60 x^{3}$ with respect to $x$ is $60 \frac{d}{d x} [x^{3}]$ .

$60 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [48 x^{2}] + \frac{d}{d x} [- 36]$

Step 3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$60 (3 x^{2}) + \frac{d}{d x} [48 x^{2}] + \frac{d}{d x} [- 36]$

Step 3.2.3

Multiply $3$ by $60$ .

$180 x^{2} + \frac{d}{d x} [48 x^{2}] + \frac{d}{d x} [- 36]$

Step 3.3

Evaluate $\frac{d}{d x} [48 x^{2}]$ .

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Step 3.3.1

Since $48$ is constant with respect to $x$ , the derivative of $48 x^{2}$ with respect to $x$ is $48 \frac{d}{d x} [x^{2}]$ .

$180 x^{2} + 48 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 36]$

Step 3.3.2

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

$180 x^{2} + 48 (2 x) + \frac{d}{d x} [- 36]$

Step 3.3.3

Multiply $2$ by $48$ .

$180 x^{2} + 96 x + \frac{d}{d x} [- 36]$

Step 3.4

Differentiate using the Constant Rule.

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Step 3.4.1

Since $- 36$ is constant with respect to $x$ , the derivative of $- 36$ with respect to $x$ is $0$ .

$180 x^{2} + 96 x + 0$

Step 3.4.2

Add $180 x^{2} + 96 x$ and $0$ .

$f''' (x) = 180 x^{2} + 96 x$