Calculus Examples

$y = \ln ({(5 x^{8} + 7 x)}^{\frac{9}{5}})$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = {(5 x^{8} + 7 x)}^{\frac{9}{5}}$ .

Tap for more steps...

Step 1.1

To apply the Chain Rule, set $u_{1}$ as ${(5 x^{8} + 7 x)}^{\frac{9}{5}}$ .

$\frac{d}{d u_{1}} [\ln (u_{1})] \frac{d}{d x} [{(5 x^{8} + 7 x)}^{\frac{9}{5}}]$

Step 1.2

The derivative of $\ln (u_{1})$ with respect to $u_{1}$ is $\frac{1}{u_{1}}$ .

$\frac{1}{u_{1}} \frac{d}{d x} [{(5 x^{8} + 7 x)}^{\frac{9}{5}}]$

Step 1.3

Replace all occurrences of $u_{1}$ with ${(5 x^{8} + 7 x)}^{\frac{9}{5}}$ .

$\frac{1}{{(5 x^{8} + 7 x)}^{\frac{9}{5}}} \frac{d}{d x} [{(5 x^{8} + 7 x)}^{\frac{9}{5}}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{9}{5}}$ and $g (x) = 5 x^{8} + 7 x$ .

Tap for more steps...

Step 2.1

To apply the Chain Rule, set $u_{2}$ as $5 x^{8} + 7 x$ .

$\frac{1}{{(5 x^{8} + 7 x)}^{\frac{9}{5}}} (\frac{d}{d u_{2}} [{u_{2}}^{\frac{9}{5}}] \frac{d}{d x} [5 x^{8} + 7 x])$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = \frac{9}{5}$ .

$\frac{1}{{(5 x^{8} + 7 x)}^{\frac{9}{5}}} (\frac{9}{5} {u_{2}}^{\frac{9}{5} - 1} \frac{d}{d x} [5 x^{8} + 7 x])$

Step 2.3

Replace all occurrences of $u_{2}$ with $5 x^{8} + 7 x$ .

$\frac{1}{{(5 x^{8} + 7 x)}^{\frac{9}{5}}} (\frac{9}{5} {(5 x^{8} + 7 x)}^{\frac{9}{5} - 1} \frac{d}{d x} [5 x^{8} + 7 x])$

Step 3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{1}{{(5 x^{8} + 7 x)}^{\frac{9}{5}}} (\frac{9}{5} {(5 x^{8} + 7 x)}^{\frac{9}{5} - 1 \cdot \frac{5}{5}} \frac{d}{d x} [5 x^{8} + 7 x])$

Step 4

Combine $- 1$ and $\frac{5}{5}$ .

$\frac{1}{{(5 x^{8} + 7 x)}^{\frac{9}{5}}} (\frac{9}{5} {(5 x^{8} + 7 x)}^{\frac{9}{5} + \frac{- 1 \cdot 5}{5}} \frac{d}{d x} [5 x^{8} + 7 x])$

Step 5

Combine the numerators over the common denominator.

$\frac{1}{{(5 x^{8} + 7 x)}^{\frac{9}{5}}} (\frac{9}{5} {(5 x^{8} + 7 x)}^{\frac{9 - 1 \cdot 5}{5}} \frac{d}{d x} [5 x^{8} + 7 x])$

Step 6

Simplify the numerator.

Tap for more steps...

Step 6.1

Multiply $- 1$ by $5$ .

$\frac{1}{{(5 x^{8} + 7 x)}^{\frac{9}{5}}} (\frac{9}{5} {(5 x^{8} + 7 x)}^{\frac{9 - 5}{5}} \frac{d}{d x} [5 x^{8} + 7 x])$

Step 6.2

Subtract $5$ from $9$ .

$\frac{1}{{(5 x^{8} + 7 x)}^{\frac{9}{5}}} (\frac{9}{5} {(5 x^{8} + 7 x)}^{\frac{4}{5}} \frac{d}{d x} [5 x^{8} + 7 x])$

Step 7

Combine $\frac{9}{5}$ and ${(5 x^{8} + 7 x)}^{\frac{4}{5}}$ .

$\frac{1}{{(5 x^{8} + 7 x)}^{\frac{9}{5}}} (\frac{9 {(5 x^{8} + 7 x)}^{\frac{4}{5}}}{5} \frac{d}{d x} [5 x^{8} + 7 x])$

Step 8

Multiply $\frac{9 {(5 x^{8} + 7 x)}^{\frac{4}{5}}}{5}$ by $\frac{1}{{(5 x^{8} + 7 x)}^{\frac{9}{5}}}$ .

$\frac{9 {(5 x^{8} + 7 x)}^{\frac{4}{5}}}{5 {(5 x^{8} + 7 x)}^{\frac{9}{5}}} \frac{d}{d x} [5 x^{8} + 7 x]$

Step 9

Move ${(5 x^{8} + 7 x)}^{\frac{4}{5}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{9}{5 {(5 x^{8} + 7 x)}^{\frac{9}{5}} {(5 x^{8} + 7 x)}^{- \frac{4}{5}}} \frac{d}{d x} [5 x^{8} + 7 x]$

Step 10

Simplify the denominator.

Tap for more steps...

Step 10.1

Multiply ${(5 x^{8} + 7 x)}^{\frac{9}{5}}$ by ${(5 x^{8} + 7 x)}^{- \frac{4}{5}}$ by adding the exponents.

Tap for more steps...

Step 10.1.1

Move ${(5 x^{8} + 7 x)}^{- \frac{4}{5}}$ .

$\frac{9}{5 ({(5 x^{8} + 7 x)}^{- \frac{4}{5}} {(5 x^{8} + 7 x)}^{\frac{9}{5}})} \frac{d}{d x} [5 x^{8} + 7 x]$

Step 10.1.2

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

$\frac{9}{5 {(5 x^{8} + 7 x)}^{- \frac{4}{5} + \frac{9}{5}}} \frac{d}{d x} [5 x^{8} + 7 x]$

Step 10.1.3

Combine the numerators over the common denominator.

$\frac{9}{5 {(5 x^{8} + 7 x)}^{\frac{- 4 + 9}{5}}} \frac{d}{d x} [5 x^{8} + 7 x]$

Step 10.1.4

Add $- 4$ and $9$ .

$\frac{9}{5 {(5 x^{8} + 7 x)}^{\frac{5}{5}}} \frac{d}{d x} [5 x^{8} + 7 x]$

Step 10.1.5

Divide $5$ by $5$ .

$\frac{9}{5 {(5 x^{8} + 7 x)}^{1}} \frac{d}{d x} [5 x^{8} + 7 x]$

Step 10.2

Simplify $5 {(5 x^{8} + 7 x)}^{1}$ .

$\frac{9}{5 (5 x^{8} + 7 x)} \frac{d}{d x} [5 x^{8} + 7 x]$

Step 11

By the Sum Rule, the derivative of $5 x^{8} + 7 x$ with respect to $x$ is $\frac{d}{d x} [5 x^{8}] + \frac{d}{d x} [7 x]$ .

$\frac{9}{5 (5 x^{8} + 7 x)} (\frac{d}{d x} [5 x^{8}] + \frac{d}{d x} [7 x])$

Step 12

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{8}$ with respect to $x$ is $5 \frac{d}{d x} [x^{8}]$ .

$\frac{9}{5 (5 x^{8} + 7 x)} (5 \frac{d}{d x} [x^{8}] + \frac{d}{d x} [7 x])$

Step 13

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 8$ .

$\frac{9}{5 (5 x^{8} + 7 x)} (5 (8 x^{7}) + \frac{d}{d x} [7 x])$

Step 14

Multiply $8$ by $5$ .

$\frac{9}{5 (5 x^{8} + 7 x)} (40 x^{7} + \frac{d}{d x} [7 x])$

Step 15

Since $7$ is constant with respect to $x$ , the derivative of $7 x$ with respect to $x$ is $7 \frac{d}{d x} [x]$ .

$\frac{9}{5 (5 x^{8} + 7 x)} (40 x^{7} + 7 \frac{d}{d x} [x])$

Step 16

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

$\frac{9}{5 (5 x^{8} + 7 x)} (40 x^{7} + 7 \cdot 1)$

Step 17

Multiply $7$ by $1$ .

$\frac{9}{5 (5 x^{8} + 7 x)} (40 x^{7} + 7)$

Step 18

Simplify.

Tap for more steps...

Step 18.1

Apply the distributive property.

$\frac{9}{5 (5 x^{8}) + 5 (7 x)} (40 x^{7} + 7)$

Step 18.2

Combine terms.

Tap for more steps...

Step 18.2.1

Multiply $5$ by $5$ .

$\frac{9}{25 x^{8} + 5 (7 x)} (40 x^{7} + 7)$

Step 18.2.2

Multiply $7$ by $5$ .

$\frac{9}{25 x^{8} + 35 x} (40 x^{7} + 7)$

Step 18.3

Reorder the factors of $\frac{9}{25 x^{8} + 35 x} (40 x^{7} + 7)$ .

$(40 x^{7} + 7) \frac{9}{25 x^{8} + 35 x}$

Step 18.4

Factor $5 x$ out of $25 x^{8} + 35 x$ .

Tap for more steps...

Step 18.4.1

Factor $5 x$ out of $25 x^{8}$ .

$(40 x^{7} + 7) \frac{9}{5 x (5 x^{7}) + 35 x}$

Step 18.4.2

Factor $5 x$ out of $35 x$ .

$(40 x^{7} + 7) \frac{9}{5 x (5 x^{7}) + 5 x (7)}$

Step 18.4.3

Factor $5 x$ out of $5 x (5 x^{7}) + 5 x (7)$ .

$(40 x^{7} + 7) \frac{9}{5 x (5 x^{7} + 7)}$

Step 18.5

Multiply $40 x^{7} + 7$ by $\frac{9}{5 x (5 x^{7} + 7)}$ .

$\frac{(40 x^{7} + 7) \cdot 9}{5 x (5 x^{7} + 7)}$

Step 18.6

Move $9$ to the left of $40 x^{7} + 7$ .

$\frac{9 (40 x^{7} + 7)}{5 x (5 x^{7} + 7)}$