Calculus Examples

$y = 10 x^{3} + 9 x^{2}$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $10 x^{3} + 9 x^{2}$ with respect to $x$ is $\frac{d}{d x} [10 x^{3}] + \frac{d}{d x} [9 x^{2}]$ .

$\frac{d}{d x} [10 x^{3}] + \frac{d}{d x} [9 x^{2}]$

Step 1.2

Evaluate $\frac{d}{d x} [10 x^{3}]$ .

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Step 1.2.1

Since $10$ is constant with respect to $x$ , the derivative of $10 x^{3}$ with respect to $x$ is $10 \frac{d}{d x} [x^{3}]$ .

$10 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [9 x^{2}]$

Step 1.2.2

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

$10 (3 x^{2}) + \frac{d}{d x} [9 x^{2}]$

Step 1.2.3

Multiply $3$ by $10$ .

$30 x^{2} + \frac{d}{d x} [9 x^{2}]$

Step 1.3

Evaluate $\frac{d}{d x} [9 x^{2}]$ .

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Step 1.3.1

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{2}$ with respect to $x$ is $9 \frac{d}{d x} [x^{2}]$ .

$30 x^{2} + 9 \frac{d}{d x} [x^{2}]$

Step 1.3.2

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

$30 x^{2} + 9 (2 x)$

Step 1.3.3

Multiply $2$ by $9$ .

$f' (x) = 30 x^{2} + 18 x$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $30 x^{2} + 18 x$ with respect to $x$ is $\frac{d}{d x} [30 x^{2}] + \frac{d}{d x} [18 x]$ .

$\frac{d}{d x} [30 x^{2}] + \frac{d}{d x} [18 x]$

Step 2.2

Evaluate $\frac{d}{d x} [30 x^{2}]$ .

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Step 2.2.1

Since $30$ is constant with respect to $x$ , the derivative of $30 x^{2}$ with respect to $x$ is $30 \frac{d}{d x} [x^{2}]$ .

$30 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [18 x]$

Step 2.2.2

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

$30 (2 x) + \frac{d}{d x} [18 x]$

Step 2.2.3

Multiply $2$ by $30$ .

$60 x + \frac{d}{d x} [18 x]$

Step 2.3

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

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Step 2.3.1

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

$60 x + 18 \frac{d}{d x} [x]$

Step 2.3.2

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

$60 x + 18 \cdot 1$

Step 2.3.3

Multiply $18$ by $1$ .

$f'' (x) = 60 x + 18$

Step 3

Find the third derivative.

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Step 3.1

By the Sum Rule, the derivative of $60 x + 18$ with respect to $x$ is $\frac{d}{d x} [60 x] + \frac{d}{d x} [18]$ .

$\frac{d}{d x} [60 x] + \frac{d}{d x} [18]$

Step 3.2

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

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Step 3.2.1

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

$60 \frac{d}{d x} [x] + \frac{d}{d x} [18]$

Step 3.2.2

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

$60 \cdot 1 + \frac{d}{d x} [18]$

Step 3.2.3

Multiply $60$ by $1$ .

$60 + \frac{d}{d x} [18]$

Step 3.3

Differentiate using the Constant Rule.

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Step 3.3.1

Since $18$ is constant with respect to $x$ , the derivative of $18$ with respect to $x$ is $0$ .

$60 + 0$

Step 3.3.2

Add $60$ and $0$ .

$f''' (x) = 60$

Step 4

Since $60$ is constant with respect to $x$ , the derivative of $60$ with respect to $x$ is $0$ .

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