Calculus Examples

$f (x) = \frac{72}{5} + \frac{48}{5 {(3 x + 1)}^{2}}$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $\frac{72}{5} + \frac{48}{5 {(3 x + 1)}^{2}}$ with respect to $x$ is $\frac{d}{d x} [\frac{72}{5}] + \frac{d}{d x} [\frac{48}{5 {(3 x + 1)}^{2}}]$ .

$\frac{d}{d x} [\frac{72}{5}] + \frac{d}{d x} [\frac{48}{5 {(3 x + 1)}^{2}}]$

Step 1.2

Since $\frac{72}{5}$ is constant with respect to $x$ , the derivative of $\frac{72}{5}$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [\frac{48}{5 {(3 x + 1)}^{2}}]$

Step 2

Evaluate $\frac{d}{d x} [\frac{48}{5 {(3 x + 1)}^{2}}]$ .

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Step 2.1

Since $\frac{48}{5}$ is constant with respect to $x$ , the derivative of $\frac{48}{5 {(3 x + 1)}^{2}}$ with respect to $x$ is $\frac{48}{5} \frac{d}{d x} [\frac{1}{{(3 x + 1)}^{2}}]$ .

$0 + \frac{48}{5} \frac{d}{d x} [\frac{1}{{(3 x + 1)}^{2}}]$

Step 2.2

Rewrite $\frac{1}{{(3 x + 1)}^{2}}$ as ${({(3 x + 1)}^{2})}^{- 1}$ .

$0 + \frac{48}{5} \frac{d}{d x} [{({(3 x + 1)}^{2})}^{- 1}]$

Step 2.3

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

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Step 2.3.1

To apply the Chain Rule, set $u_{1}$ as ${(3 x + 1)}^{2}$ .

$0 + \frac{48}{5} (\frac{d}{d u_{1}} [{u_{1}}^{- 1}] \frac{d}{d x} [{(3 x + 1)}^{2}])$

Step 2.3.2

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

$0 + \frac{48}{5} (- {u_{1}}^{- 2} \frac{d}{d x} [{(3 x + 1)}^{2}])$

Step 2.3.3

Replace all occurrences of $u_{1}$ with ${(3 x + 1)}^{2}$ .

$0 + \frac{48}{5} (- {({(3 x + 1)}^{2})}^{- 2} \frac{d}{d x} [{(3 x + 1)}^{2}])$

Step 2.4

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

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Step 2.4.1

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

$0 + \frac{48}{5} (- {({(3 x + 1)}^{2})}^{- 2} (\frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d x} [3 x + 1]))$

Step 2.4.2

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

$0 + \frac{48}{5} (- {({(3 x + 1)}^{2})}^{- 2} (2 u_{2} \frac{d}{d x} [3 x + 1]))$

Step 2.4.3

Replace all occurrences of $u_{2}$ with $3 x + 1$ .

$0 + \frac{48}{5} (- {({(3 x + 1)}^{2})}^{- 2} (2 (3 x + 1) \frac{d}{d x} [3 x + 1]))$

Step 2.5

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

$0 + \frac{48}{5} (- {({(3 x + 1)}^{2})}^{- 2} (2 (3 x + 1) (\frac{d}{d x} [3 x] + \frac{d}{d x} [1])))$

Step 2.6

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

$0 + \frac{48}{5} (- {({(3 x + 1)}^{2})}^{- 2} (2 (3 x + 1) (3 \frac{d}{d x} [x] + \frac{d}{d x} [1])))$

Step 2.7

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

$0 + \frac{48}{5} (- {({(3 x + 1)}^{2})}^{- 2} (2 (3 x + 1) (3 \cdot 1 + \frac{d}{d x} [1])))$

Step 2.8

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

$0 + \frac{48}{5} (- {({(3 x + 1)}^{2})}^{- 2} (2 (3 x + 1) (3 \cdot 1 + 0)))$

Step 2.9

Multiply the exponents in ${({(3 x + 1)}^{2})}^{- 2}$ .

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Step 2.9.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$0 + \frac{48}{5} (- {(3 x + 1)}^{2 \cdot - 2} (2 (3 x + 1) (3 \cdot 1 + 0)))$

Step 2.9.2

Multiply $2$ by $- 2$ .

$0 + \frac{48}{5} (- {(3 x + 1)}^{- 4} (2 (3 x + 1) (3 \cdot 1 + 0)))$

Step 2.10

Multiply $3$ by $1$ .

$0 + \frac{48}{5} (- {(3 x + 1)}^{- 4} (2 (3 x + 1) (3 + 0)))$

Step 2.11

Add $3$ and $0$ .

$0 + \frac{48}{5} (- {(3 x + 1)}^{- 4} (2 (3 x + 1) \cdot 3))$

Step 2.12

Multiply $3$ by $2$ .

$0 + \frac{48}{5} (- {(3 x + 1)}^{- 4} (6 (3 x + 1)))$

Step 2.13

Multiply $6$ by $- 1$ .

$0 + \frac{48}{5} (- 6 {(3 x + 1)}^{- 4} (3 x + 1))$

Step 2.14

Raise $3 x + 1$ to the power of $1$ .

$0 + \frac{48}{5} (- 6 ({(3 x + 1)}^{1} {(3 x + 1)}^{- 4}))$

Step 2.15

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

$0 + \frac{48}{5} (- 6 {(3 x + 1)}^{1 - 4})$

Step 2.16

Subtract $4$ from $1$ .

$0 + \frac{48}{5} (- 6 {(3 x + 1)}^{- 3})$

Step 2.17

Combine $- 6$ and $\frac{48}{5}$ .

$0 + \frac{- 6 \cdot 48}{5} {(3 x + 1)}^{- 3}$

Step 2.18

Multiply $- 6$ by $48$ .

$0 + \frac{- 288}{5} {(3 x + 1)}^{- 3}$

Step 2.19

Combine $\frac{- 288}{5}$ and ${(3 x + 1)}^{- 3}$ .

$0 + \frac{- 288 {(3 x + 1)}^{- 3}}{5}$

Step 2.20

Move ${(3 x + 1)}^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$0 + \frac{- 288}{5 {(3 x + 1)}^{3}}$

Step 2.21

Move the negative in front of the fraction.

$0 - \frac{288}{5 {(3 x + 1)}^{3}}$

Step 3

Subtract $\frac{288}{5 {(3 x + 1)}^{3}}$ from $0$ .

$- \frac{288}{5 {(3 x + 1)}^{3}}$