Calculus Examples

$y = {(3 x^{2} + 5 x + 1)}^{\frac{3}{2}}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{3}{2}}$ and $g (x) = 3 x^{2} + 5 x + 1$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $3 x^{2} + 5 x + 1$ .

$\frac{d}{d u} [u^{\frac{3}{2}}] \frac{d}{d x} [3 x^{2} + 5 x + 1]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{3}{2}$ .

$\frac{3}{2} u^{\frac{3}{2} - 1} \frac{d}{d x} [3 x^{2} + 5 x + 1]$

Step 1.3

Replace all occurrences of $u$ with $3 x^{2} + 5 x + 1$ .

$\frac{3}{2} {(3 x^{2} + 5 x + 1)}^{\frac{3}{2} - 1} \frac{d}{d x} [3 x^{2} + 5 x + 1]$

Step 2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3}{2} {(3 x^{2} + 5 x + 1)}^{\frac{3}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [3 x^{2} + 5 x + 1]$

Step 3

Combine $- 1$ and $\frac{2}{2}$ .

$\frac{3}{2} {(3 x^{2} + 5 x + 1)}^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [3 x^{2} + 5 x + 1]$

Step 4

Combine the numerators over the common denominator.

$\frac{3}{2} {(3 x^{2} + 5 x + 1)}^{\frac{3 - 1 \cdot 2}{2}} \frac{d}{d x} [3 x^{2} + 5 x + 1]$

Step 5

Simplify the numerator.

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Step 5.1

Multiply $- 1$ by $2$ .

$\frac{3}{2} {(3 x^{2} + 5 x + 1)}^{\frac{3 - 2}{2}} \frac{d}{d x} [3 x^{2} + 5 x + 1]$

Step 5.2

Subtract $2$ from $3$ .

$\frac{3}{2} {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}} \frac{d}{d x} [3 x^{2} + 5 x + 1]$

Step 6

Combine $\frac{3}{2}$ and ${(3 x^{2} + 5 x + 1)}^{\frac{1}{2}}$ .

$\frac{3 {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}}}{2} \frac{d}{d x} [3 x^{2} + 5 x + 1]$

Step 7

By the Sum Rule, the derivative of $3 x^{2} + 5 x + 1$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [1]$ .

$\frac{3 {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}}}{2} (\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [1])$

Step 8

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

$\frac{3 {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}}}{2} (3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [1])$

Step 9

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

$\frac{3 {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}}}{2} (3 (2 x) + \frac{d}{d x} [5 x] + \frac{d}{d x} [1])$

Step 10

Multiply $2$ by $3$ .

$\frac{3 {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}}}{2} (6 x + \frac{d}{d x} [5 x] + \frac{d}{d x} [1])$

Step 11

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

$\frac{3 {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}}}{2} (6 x + 5 \frac{d}{d x} [x] + \frac{d}{d x} [1])$

Step 12

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

$\frac{3 {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}}}{2} (6 x + 5 \cdot 1 + \frac{d}{d x} [1])$

Step 13

Multiply $5$ by $1$ .

$\frac{3 {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}}}{2} (6 x + 5 + \frac{d}{d x} [1])$

Step 14

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{3 {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}}}{2} (6 x + 5 + 0)$

Step 15

Add $6 x + 5$ and $0$ .

$\frac{3 {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}}}{2} (6 x + 5)$

Step 16

Simplify.

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Step 16.1

Reorder the factors of $\frac{3 {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}}}{2} (6 x + 5)$ .

$(6 x + 5) \frac{3 {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}}}{2}$

Step 16.2

Apply the distributive property.

$6 x \frac{3 {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}}}{2} + 5 \frac{3 {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}}}{2}$

Step 16.3

Cancel the common factor of $2$ .

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Step 16.3.1

Factor $2$ out of $6 x$ .

$2 (3 x) \frac{3 {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}}}{2} + 5 \frac{3 {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}}}{2}$

Step 16.3.2

Cancel the common factor.

$2 (3 x) \frac{3 {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}}}{2} + 5 \frac{3 {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}}}{2}$

Step 16.3.3

Rewrite the expression.

$3 x (3 {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}}) + 5 \frac{3 {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}}}{2}$

Step 16.4

Multiply $3$ by $3$ .

$9 x {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}} + 5 \frac{3 {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}}}{2}$

Step 16.5

Multiply $5 \frac{3 {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}}}{2}$ .

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Step 16.5.1

Combine $5$ and $\frac{3 {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}}}{2}$ .

$9 x {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}} + \frac{5 (3 {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}})}{2}$

Step 16.5.2

Multiply $3$ by $5$ .

$9 x {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}} + \frac{15 {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}}}{2}$

Step 16.6

To write $9 x {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$9 x {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}} \cdot \frac{2}{2} + \frac{15 {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}}}{2}$

Step 16.7

Combine $9 x {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}}$ and $\frac{2}{2}$ .

$\frac{9 x {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}} \cdot 2}{2} + \frac{15 {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}}}{2}$

Step 16.8

Combine the numerators over the common denominator.

$\frac{9 x {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}} \cdot 2 + 15 {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}}}{2}$

Step 16.9

Simplify the numerator.

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Step 16.9.1

Factor $3 {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}}$ out of $9 x {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}} \cdot 2 + 15 {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}}$ .

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Step 16.9.1.1

Reorder the expression.

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Step 16.9.1.1.1

Move ${(3 x^{2} + 5 x + 1)}^{\frac{1}{2}}$ .

$\frac{9 x \cdot 2 {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}} + 15 {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}}}{2}$

Step 16.9.1.1.2

Move $x$ .

$\frac{9 \cdot 2 x {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}} + 15 {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}}}{2}$

Step 16.9.1.2

Factor $3 {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}}$ out of $9 \cdot 2 x {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}}$ .

$\frac{3 {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}} (3 \cdot 2 x) + 15 {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}}}{2}$

Step 16.9.1.3

Factor $3 {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}}$ out of $15 {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}}$ .

$\frac{3 {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}} (3 \cdot 2 x) + 3 {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}} (5)}{2}$

Step 16.9.1.4

Factor $3 {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}}$ out of $3 {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}} (3 \cdot 2 x) + 3 {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}} (5)$ .

$\frac{3 {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}} (3 \cdot 2 x + 5)}{2}$

Step 16.9.2

Multiply $3$ by $2$ .

$\frac{3 {(3 x^{2} + 5 x + 1)}^{\frac{1}{2}} (6 x + 5)}{2}$