Calculus Examples

$y = \frac{7}{4} \cdot \sqrt[6]{9 - 8 x^{16}}$

Step 1

Differentiate using the Constant Multiple Rule.

Tap for more steps...

Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[6]{9 - 8 x^{16}}$ as ${(9 - 8 x^{16})}^{\frac{1}{6}}$ .

$\frac{d}{d x} [\frac{7}{4} \cdot {(9 - 8 x^{16})}^{\frac{1}{6}}]$

Step 1.2

Since $\frac{7}{4}$ is constant with respect to $x$ , the derivative of $\frac{7}{4} \cdot {(9 - 8 x^{16})}^{\frac{1}{6}}$ with respect to $x$ is $\frac{7}{4} \frac{d}{d x} [{(9 - 8 x^{16})}^{\frac{1}{6}}]$ .

$\frac{7}{4} \frac{d}{d x} [{(9 - 8 x^{16})}^{\frac{1}{6}}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{6}}$ and $g (x) = 9 - 8 x^{16}$ .

Tap for more steps...

Step 2.1

To apply the Chain Rule, set $u$ as $9 - 8 x^{16}$ .

$\frac{7}{4} (\frac{d}{d u} [u^{\frac{1}{6}}] \frac{d}{d x} [9 - 8 x^{16}])$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{1}{6}$ .

$\frac{7}{4} (\frac{1}{6} u^{\frac{1}{6} - 1} \frac{d}{d x} [9 - 8 x^{16}])$

Step 2.3

Replace all occurrences of $u$ with $9 - 8 x^{16}$ .

$\frac{7}{4} (\frac{1}{6} {(9 - 8 x^{16})}^{\frac{1}{6} - 1} \frac{d}{d x} [9 - 8 x^{16}])$

Step 3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{7}{4} (\frac{1}{6} {(9 - 8 x^{16})}^{\frac{1}{6} - 1 \cdot \frac{6}{6}} \frac{d}{d x} [9 - 8 x^{16}])$

Step 4

Combine $- 1$ and $\frac{6}{6}$ .

$\frac{7}{4} (\frac{1}{6} {(9 - 8 x^{16})}^{\frac{1}{6} + \frac{- 1 \cdot 6}{6}} \frac{d}{d x} [9 - 8 x^{16}])$

Step 5

Combine the numerators over the common denominator.

$\frac{7}{4} (\frac{1}{6} {(9 - 8 x^{16})}^{\frac{1 - 1 \cdot 6}{6}} \frac{d}{d x} [9 - 8 x^{16}])$

Step 6

Simplify the numerator.

Tap for more steps...

Step 6.1

Multiply $- 1$ by $6$ .

$\frac{7}{4} (\frac{1}{6} {(9 - 8 x^{16})}^{\frac{1 - 6}{6}} \frac{d}{d x} [9 - 8 x^{16}])$

Step 6.2

Subtract $6$ from $1$ .

$\frac{7}{4} (\frac{1}{6} {(9 - 8 x^{16})}^{\frac{- 5}{6}} \frac{d}{d x} [9 - 8 x^{16}])$

Step 7

Combine fractions.

Tap for more steps...

Step 7.1

Move the negative in front of the fraction.

$\frac{7}{4} (\frac{1}{6} {(9 - 8 x^{16})}^{- \frac{5}{6}} \frac{d}{d x} [9 - 8 x^{16}])$

Step 7.2

Combine $\frac{1}{6}$ and ${(9 - 8 x^{16})}^{- \frac{5}{6}}$ .

$\frac{7}{4} (\frac{{(9 - 8 x^{16})}^{- \frac{5}{6}}}{6} \frac{d}{d x} [9 - 8 x^{16}])$

Step 7.3

Move ${(9 - 8 x^{16})}^{- \frac{5}{6}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{7}{4} (\frac{1}{6 {(9 - 8 x^{16})}^{\frac{5}{6}}} \frac{d}{d x} [9 - 8 x^{16}])$

Step 7.4

Multiply $\frac{1}{6 {(9 - 8 x^{16})}^{\frac{5}{6}}}$ by $\frac{7}{4}$ .

$\frac{7}{6 {(9 - 8 x^{16})}^{\frac{5}{6}} \cdot 4} \frac{d}{d x} [9 - 8 x^{16}]$

Step 7.5

Multiply $4$ by $6$ .

$\frac{7}{24 {(9 - 8 x^{16})}^{\frac{5}{6}}} \frac{d}{d x} [9 - 8 x^{16}]$

Step 8

By the Sum Rule, the derivative of $9 - 8 x^{16}$ with respect to $x$ is $\frac{d}{d x} [9] + \frac{d}{d x} [- 8 x^{16}]$ .

$\frac{7}{24 {(9 - 8 x^{16})}^{\frac{5}{6}}} (\frac{d}{d x} [9] + \frac{d}{d x} [- 8 x^{16}])$

Step 9

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$\frac{7}{24 {(9 - 8 x^{16})}^{\frac{5}{6}}} (0 + \frac{d}{d x} [- 8 x^{16}])$

Step 10

Add $0$ and $\frac{d}{d x} [- 8 x^{16}]$ .

$\frac{7}{24 {(9 - 8 x^{16})}^{\frac{5}{6}}} \frac{d}{d x} [- 8 x^{16}]$

Step 11

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 x^{16}$ with respect to $x$ is $- 8 \frac{d}{d x} [x^{16}]$ .

$\frac{7}{24 {(9 - 8 x^{16})}^{\frac{5}{6}}} (- 8 \frac{d}{d x} [x^{16}])$

Step 12

Simplify terms.

Tap for more steps...

Step 12.1

Combine $- 8$ and $\frac{7}{24 {(9 - 8 x^{16})}^{\frac{5}{6}}}$ .

$\frac{- 8 \cdot 7}{24 {(9 - 8 x^{16})}^{\frac{5}{6}}} \frac{d}{d x} [x^{16}]$

Step 12.2

Multiply $- 8$ by $7$ .

$\frac{- 56}{24 {(9 - 8 x^{16})}^{\frac{5}{6}}} \frac{d}{d x} [x^{16}]$

Step 12.3

Factor $8$ out of $- 56$ .

$\frac{8 (- 7)}{24 {(9 - 8 x^{16})}^{\frac{5}{6}}} \frac{d}{d x} [x^{16}]$

Step 13

Cancel the common factors.

Tap for more steps...

Step 13.1

Factor $8$ out of $24 {(9 - 8 x^{16})}^{\frac{5}{6}}$ .

$\frac{8 (- 7)}{8 (3 {(9 - 8 x^{16})}^{\frac{5}{6}})} \frac{d}{d x} [x^{16}]$

Step 13.2

Cancel the common factor.

$\frac{8 \cdot - 7}{8 (3 {(9 - 8 x^{16})}^{\frac{5}{6}})} \frac{d}{d x} [x^{16}]$

Step 13.3

Rewrite the expression.

$\frac{- 7}{3 {(9 - 8 x^{16})}^{\frac{5}{6}}} \frac{d}{d x} [x^{16}]$

Step 14

Move the negative in front of the fraction.

$- \frac{7}{3 {(9 - 8 x^{16})}^{\frac{5}{6}}} \frac{d}{d x} [x^{16}]$

Step 15

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 16$ .

$- \frac{7}{3 {(9 - 8 x^{16})}^{\frac{5}{6}}} (16 x^{15})$

Step 16

Combine fractions.

Tap for more steps...

Step 16.1

Multiply $16$ by $- 1$ .

$- 16 \frac{7}{3 {(9 - 8 x^{16})}^{\frac{5}{6}}} x^{15}$

Step 16.2

Combine $- 16$ and $\frac{7}{3 {(9 - 8 x^{16})}^{\frac{5}{6}}}$ .

$\frac{- 16 \cdot 7}{3 {(9 - 8 x^{16})}^{\frac{5}{6}}} x^{15}$

Step 16.3

Multiply $- 16$ by $7$ .

$\frac{- 112}{3 {(9 - 8 x^{16})}^{\frac{5}{6}}} x^{15}$

Step 16.4

Combine $\frac{- 112}{3 {(9 - 8 x^{16})}^{\frac{5}{6}}}$ and $x^{15}$ .

$\frac{- 112 x^{15}}{3 {(9 - 8 x^{16})}^{\frac{5}{6}}}$

Step 16.5

Move the negative in front of the fraction.

$- \frac{112 x^{15}}{3 {(9 - 8 x^{16})}^{\frac{5}{6}}}$