Calculus Examples

$y = 5 x^{3} - 3 x^{5}$

Step 1

Find the first derivative.

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Step 1.1

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

$\frac{d}{d x} [5 x^{3}] + \frac{d}{d x} [- 3 x^{5}]$

Step 1.2

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

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Step 1.2.1

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

$5 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 3 x^{5}]$

Step 1.2.2

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

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

Step 1.2.3

Multiply $3$ by $5$ .

$15 x^{2} + \frac{d}{d x} [- 3 x^{5}]$

Step 1.3

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

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Step 1.3.1

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

$15 x^{2} - 3 \frac{d}{d x} [x^{5}]$

Step 1.3.2

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

$15 x^{2} - 3 (5 x^{4})$

Step 1.3.3

Multiply $5$ by $- 3$ .

$15 x^{2} - 15 x^{4}$

Step 1.4

Reorder terms.

$f' (x) = - 15 x^{4} + 15 x^{2}$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $- 15 x^{4} + 15 x^{2}$ with respect to $x$ is $\frac{d}{d x} [- 15 x^{4}] + \frac{d}{d x} [15 x^{2}]$ .

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

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

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

Step 2.2.2

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

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

Step 2.2.3

Multiply $4$ by $- 15$ .

$- 60 x^{3} + \frac{d}{d x} [15 x^{2}]$

Step 2.3

Evaluate $\frac{d}{d x} [15 x^{2}]$ .

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Step 2.3.1

Since $15$ is constant with respect to $x$ , the derivative of $15 x^{2}$ with respect to $x$ is $15 \frac{d}{d x} [x^{2}]$ .

$- 60 x^{3} + 15 \frac{d}{d x} [x^{2}]$

Step 2.3.2

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

$- 60 x^{3} + 15 (2 x)$

Step 2.3.3

Multiply $2$ by $15$ .

$f'' (x) = - 60 x^{3} + 30 x$

Step 3

Find the third derivative.

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Step 3.1

By the Sum Rule, the derivative of $- 60 x^{3} + 30 x$ with respect to $x$ is $\frac{d}{d x} [- 60 x^{3}] + \frac{d}{d x} [30 x]$ .

$\frac{d}{d x} [- 60 x^{3}] + \frac{d}{d x} [30 x]$

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

$- 60 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [30 x]$

Step 3.2.2

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

$- 60 (3 x^{2}) + \frac{d}{d x} [30 x]$

Step 3.2.3

Multiply $3$ by $- 60$ .

$- 180 x^{2} + \frac{d}{d x} [30 x]$

Step 3.3

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

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Step 3.3.1

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

$- 180 x^{2} + 30 \frac{d}{d x} [x]$

Step 3.3.2

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

$- 180 x^{2} + 30 \cdot 1$

Step 3.3.3

Multiply $30$ by $1$ .

$f''' (x) = - 180 x^{2} + 30$

Step 4

Find the fourth derivative.

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Step 4.1

By the Sum Rule, the derivative of $- 180 x^{2} + 30$ with respect to $x$ is $\frac{d}{d x} [- 180 x^{2}] + \frac{d}{d x} [30]$ .

$\frac{d}{d x} [- 180 x^{2}] + \frac{d}{d x} [30]$

Step 4.2

Evaluate $\frac{d}{d x} [- 180 x^{2}]$ .

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Step 4.2.1

Since $- 180$ is constant with respect to $x$ , the derivative of $- 180 x^{2}$ with respect to $x$ is $- 180 \frac{d}{d x} [x^{2}]$ .

$- 180 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [30]$

Step 4.2.2

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

$- 180 (2 x) + \frac{d}{d x} [30]$

Step 4.2.3

Multiply $2$ by $- 180$ .

$- 360 x + \frac{d}{d x} [30]$

Step 4.3

Differentiate using the Constant Rule.

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Step 4.3.1

Since $30$ is constant with respect to $x$ , the derivative of $30$ with respect to $x$ is $0$ .

$- 360 x + 0$

Step 4.3.2

Add $- 360 x$ and $0$ .

$f^{4} (x) = - 360 x$