Calculus Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

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

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Multiply $2$ by $3$ .

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

Step 3.5

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

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

Step 3.6

Simplify the expression.

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Step 3.6.1

Add $6 x$ and $0$ .

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

Step 3.6.2

Move $6$ to the left of $2 x + 5$ .

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

Step 3.7

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

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

Step 3.8

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

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

Step 3.9

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

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

Step 3.10

Multiply $2$ by $1$ .

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

Step 3.11

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

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

Step 3.12

Combine fractions.

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Step 3.12.1

Add $2$ and $0$ .

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

Step 3.12.2

Multiply $2$ by $- 1$ .

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

Step 3.12.3

Combine $\frac{6 (2 x + 5) x - 2 (3 x^{2} - 1)}{{(2 x + 5)}^{2}}$ and $3$ .

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

Step 3.12.4

Move $3$ to the left of $6 (2 x + 5) x - 2 (3 x^{2} - 1)$ .

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

Step 4

Simplify.

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Step 4.1

Apply the product rule to $\frac{3 x^{2} - 1}{2 x + 5}$ .

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Apply the distributive property.

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

Step 4.4

Apply the distributive property.

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

Step 4.5

Apply the distributive property.

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

Step 4.6

Combine terms.

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Step 4.6.1

Multiply $2$ by $6$ .

$\frac{3 (12 x \cdot x) + 3 (6 \cdot 5 x) + 3 (- 2 (3 x^{2})) + 3 (- 2 \cdot - 1)}{{(2 x + 5)}^{2}} \cdot \frac{{(3 x^{2} - 1)}^{2}}{{(2 x + 5)}^{2}}$

Step 4.6.2

Raise $x$ to the power of $1$ .

$\frac{3 (12 (x^{1} x)) + 3 (6 \cdot 5 x) + 3 (- 2 (3 x^{2})) + 3 (- 2 \cdot - 1)}{{(2 x + 5)}^{2}} \cdot \frac{{(3 x^{2} - 1)}^{2}}{{(2 x + 5)}^{2}}$

Step 4.6.3

Raise $x$ to the power of $1$ .

$\frac{3 (12 (x^{1} x^{1})) + 3 (6 \cdot 5 x) + 3 (- 2 (3 x^{2})) + 3 (- 2 \cdot - 1)}{{(2 x + 5)}^{2}} \cdot \frac{{(3 x^{2} - 1)}^{2}}{{(2 x + 5)}^{2}}$

Step 4.6.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{3 (12 x^{1 + 1}) + 3 (6 \cdot 5 x) + 3 (- 2 (3 x^{2})) + 3 (- 2 \cdot - 1)}{{(2 x + 5)}^{2}} \cdot \frac{{(3 x^{2} - 1)}^{2}}{{(2 x + 5)}^{2}}$

Step 4.6.5

Add $1$ and $1$ .

$\frac{3 (12 x^{2}) + 3 (6 \cdot 5 x) + 3 (- 2 (3 x^{2})) + 3 (- 2 \cdot - 1)}{{(2 x + 5)}^{2}} \cdot \frac{{(3 x^{2} - 1)}^{2}}{{(2 x + 5)}^{2}}$

Step 4.6.6

Multiply $12$ by $3$ .

$\frac{36 x^{2} + 3 (6 \cdot 5 x) + 3 (- 2 (3 x^{2})) + 3 (- 2 \cdot - 1)}{{(2 x + 5)}^{2}} \cdot \frac{{(3 x^{2} - 1)}^{2}}{{(2 x + 5)}^{2}}$

Step 4.6.7

Multiply $6$ by $5$ .

$\frac{36 x^{2} + 3 (30 x) + 3 (- 2 (3 x^{2})) + 3 (- 2 \cdot - 1)}{{(2 x + 5)}^{2}} \cdot \frac{{(3 x^{2} - 1)}^{2}}{{(2 x + 5)}^{2}}$

Step 4.6.8

Multiply $30$ by $3$ .

$\frac{36 x^{2} + 90 x + 3 (- 2 (3 x^{2})) + 3 (- 2 \cdot - 1)}{{(2 x + 5)}^{2}} \cdot \frac{{(3 x^{2} - 1)}^{2}}{{(2 x + 5)}^{2}}$

Step 4.6.9

Multiply $3$ by $- 2$ .

$\frac{36 x^{2} + 90 x + 3 (- 6 x^{2}) + 3 (- 2 \cdot - 1)}{{(2 x + 5)}^{2}} \cdot \frac{{(3 x^{2} - 1)}^{2}}{{(2 x + 5)}^{2}}$

Step 4.6.10

Multiply $- 6$ by $3$ .

$\frac{36 x^{2} + 90 x - 18 x^{2} + 3 (- 2 \cdot - 1)}{{(2 x + 5)}^{2}} \cdot \frac{{(3 x^{2} - 1)}^{2}}{{(2 x + 5)}^{2}}$

Step 4.6.11

Multiply $- 2$ by $- 1$ .

$\frac{36 x^{2} + 90 x - 18 x^{2} + 3 \cdot 2}{{(2 x + 5)}^{2}} \cdot \frac{{(3 x^{2} - 1)}^{2}}{{(2 x + 5)}^{2}}$

Step 4.6.12

Multiply $3$ by $2$ .

$\frac{36 x^{2} + 90 x - 18 x^{2} + 6}{{(2 x + 5)}^{2}} \cdot \frac{{(3 x^{2} - 1)}^{2}}{{(2 x + 5)}^{2}}$

Step 4.6.13

Subtract $18 x^{2}$ from $36 x^{2}$ .

$\frac{18 x^{2} + 90 x + 6}{{(2 x + 5)}^{2}} \cdot \frac{{(3 x^{2} - 1)}^{2}}{{(2 x + 5)}^{2}}$

Step 4.6.14

Multiply $\frac{18 x^{2} + 90 x + 6}{{(2 x + 5)}^{2}}$ by $\frac{{(3 x^{2} - 1)}^{2}}{{(2 x + 5)}^{2}}$ .

$\frac{(18 x^{2} + 90 x + 6) {(3 x^{2} - 1)}^{2}}{{(2 x + 5)}^{2} {(2 x + 5)}^{2}}$

Step 4.6.15

Multiply ${(2 x + 5)}^{2}$ by ${(2 x + 5)}^{2}$ by adding the exponents.

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Step 4.6.15.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{(18 x^{2} + 90 x + 6) {(3 x^{2} - 1)}^{2}}{{(2 x + 5)}^{2 + 2}}$

Step 4.6.15.2

Add $2$ and $2$ .

$\frac{(18 x^{2} + 90 x + 6) {(3 x^{2} - 1)}^{2}}{{(2 x + 5)}^{4}}$

Step 4.7

Reorder terms.

$\frac{{(3 x^{2} - 1)}^{2} (18 x^{2} + 90 x + 6)}{{(2 x + 5)}^{4}}$

Step 4.8

Factor $6$ out of $18 x^{2} + 90 x + 6$ .

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Step 4.8.1

Factor $6$ out of $18 x^{2}$ .

$\frac{{(3 x^{2} - 1)}^{2} (6 (3 x^{2}) + 90 x + 6)}{{(2 x + 5)}^{4}}$

Step 4.8.2

Factor $6$ out of $90 x$ .

$\frac{{(3 x^{2} - 1)}^{2} (6 (3 x^{2}) + 6 (15 x) + 6)}{{(2 x + 5)}^{4}}$

Step 4.8.3

Factor $6$ out of $6$ .

$\frac{{(3 x^{2} - 1)}^{2} (6 (3 x^{2}) + 6 (15 x) + 6 (1))}{{(2 x + 5)}^{4}}$

Step 4.8.4

Factor $6$ out of $6 (3 x^{2}) + 6 (15 x)$ .

$\frac{{(3 x^{2} - 1)}^{2} (6 (3 x^{2} + 15 x) + 6 (1))}{{(2 x + 5)}^{4}}$

Step 4.8.5

Factor $6$ out of $6 (3 x^{2} + 15 x) + 6 (1)$ .

$\frac{{(3 x^{2} - 1)}^{2} (6 (3 x^{2} + 15 x + 1))}{{(2 x + 5)}^{4}}$

$\frac{{(3 x^{2} - 1)}^{2} \cdot 6 (3 x^{2} + 15 x + 1)}{{(2 x + 5)}^{4}}$

Step 4.9

Move $6$ to the left of ${(3 x^{2} - 1)}^{2}$ .

$\frac{6 {(3 x^{2} - 1)}^{2} (3 x^{2} + 15 x + 1)}{{(2 x + 5)}^{4}}$