Calculus Examples

$f (x) = x^{8} \sqrt{5 - 3 x}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 - 3 x}$ as ${(5 - 3 x)}^{\frac{1}{2}}$ .

$\frac{d}{d x} [x^{8} {(5 - 3 x)}^{\frac{1}{2}}]$

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{8}$ and $g (x) = {(5 - 3 x)}^{\frac{1}{2}}$ .

$x^{8} \frac{d}{d x} [{(5 - 3 x)}^{\frac{1}{2}}] + {(5 - 3 x)}^{\frac{1}{2}} \frac{d}{d x} [x^{8}]$

Step 3

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

Tap for more steps...

Step 3.1

To apply the Chain Rule, set $u$ as $5 - 3 x$ .

$x^{8} (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [5 - 3 x]) + {(5 - 3 x)}^{\frac{1}{2}} \frac{d}{d x} [x^{8}]$

Step 3.2

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

$x^{8} (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [5 - 3 x]) + {(5 - 3 x)}^{\frac{1}{2}} \frac{d}{d x} [x^{8}]$

Step 3.3

Replace all occurrences of $u$ with $5 - 3 x$ .

$x^{8} (\frac{1}{2} {(5 - 3 x)}^{\frac{1}{2} - 1} \frac{d}{d x} [5 - 3 x]) + {(5 - 3 x)}^{\frac{1}{2}} \frac{d}{d x} [x^{8}]$

Step 4

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

$x^{8} (\frac{1}{2} {(5 - 3 x)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [5 - 3 x]) + {(5 - 3 x)}^{\frac{1}{2}} \frac{d}{d x} [x^{8}]$

Step 5

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

$x^{8} (\frac{1}{2} {(5 - 3 x)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [5 - 3 x]) + {(5 - 3 x)}^{\frac{1}{2}} \frac{d}{d x} [x^{8}]$

Step 6

Combine the numerators over the common denominator.

$x^{8} (\frac{1}{2} {(5 - 3 x)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [5 - 3 x]) + {(5 - 3 x)}^{\frac{1}{2}} \frac{d}{d x} [x^{8}]$

Step 7

Simplify the numerator.

Tap for more steps...

Step 7.1

Multiply $- 1$ by $2$ .

$x^{8} (\frac{1}{2} {(5 - 3 x)}^{\frac{1 - 2}{2}} \frac{d}{d x} [5 - 3 x]) + {(5 - 3 x)}^{\frac{1}{2}} \frac{d}{d x} [x^{8}]$

Step 7.2

Subtract $2$ from $1$ .

$x^{8} (\frac{1}{2} {(5 - 3 x)}^{\frac{- 1}{2}} \frac{d}{d x} [5 - 3 x]) + {(5 - 3 x)}^{\frac{1}{2}} \frac{d}{d x} [x^{8}]$

Step 8

Combine fractions.

Tap for more steps...

Step 8.1

Move the negative in front of the fraction.

$x^{8} (\frac{1}{2} {(5 - 3 x)}^{- \frac{1}{2}} \frac{d}{d x} [5 - 3 x]) + {(5 - 3 x)}^{\frac{1}{2}} \frac{d}{d x} [x^{8}]$

Step 8.2

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

$x^{8} (\frac{{(5 - 3 x)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [5 - 3 x]) + {(5 - 3 x)}^{\frac{1}{2}} \frac{d}{d x} [x^{8}]$

Step 8.3

Move ${(5 - 3 x)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x^{8} (\frac{1}{2 {(5 - 3 x)}^{\frac{1}{2}}} \frac{d}{d x} [5 - 3 x]) + {(5 - 3 x)}^{\frac{1}{2}} \frac{d}{d x} [x^{8}]$

Step 8.4

Combine $\frac{1}{2 {(5 - 3 x)}^{\frac{1}{2}}}$ and $x^{8}$ .

$\frac{x^{8}}{2 {(5 - 3 x)}^{\frac{1}{2}}} \frac{d}{d x} [5 - 3 x] + {(5 - 3 x)}^{\frac{1}{2}} \frac{d}{d x} [x^{8}]$

Step 9

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

$\frac{x^{8}}{2 {(5 - 3 x)}^{\frac{1}{2}}} (\frac{d}{d x} [5] + \frac{d}{d x} [- 3 x]) + {(5 - 3 x)}^{\frac{1}{2}} \frac{d}{d x} [x^{8}]$

Step 10

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

$\frac{x^{8}}{2 {(5 - 3 x)}^{\frac{1}{2}}} (0 + \frac{d}{d x} [- 3 x]) + {(5 - 3 x)}^{\frac{1}{2}} \frac{d}{d x} [x^{8}]$

Step 11

Add $0$ and $\frac{d}{d x} [- 3 x]$ .

$\frac{x^{8}}{2 {(5 - 3 x)}^{\frac{1}{2}}} \frac{d}{d x} [- 3 x] + {(5 - 3 x)}^{\frac{1}{2}} \frac{d}{d x} [x^{8}]$

Step 12

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

$\frac{x^{8}}{2 {(5 - 3 x)}^{\frac{1}{2}}} (- 3 \frac{d}{d x} [x]) + {(5 - 3 x)}^{\frac{1}{2}} \frac{d}{d x} [x^{8}]$

Step 13

Combine fractions.

Tap for more steps...

Step 13.1

Combine $- 3$ and $\frac{x^{8}}{2 {(5 - 3 x)}^{\frac{1}{2}}}$ .

$\frac{- 3 x^{8}}{2 {(5 - 3 x)}^{\frac{1}{2}}} \frac{d}{d x} [x] + {(5 - 3 x)}^{\frac{1}{2}} \frac{d}{d x} [x^{8}]$

Step 13.2

Move the negative in front of the fraction.

$- \frac{3 x^{8}}{2 {(5 - 3 x)}^{\frac{1}{2}}} \frac{d}{d x} [x] + {(5 - 3 x)}^{\frac{1}{2}} \frac{d}{d x} [x^{8}]$

Step 14

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

$- \frac{3 x^{8}}{2 {(5 - 3 x)}^{\frac{1}{2}}} \cdot 1 + {(5 - 3 x)}^{\frac{1}{2}} \frac{d}{d x} [x^{8}]$

Step 15

Multiply $- 1$ by $1$ .

$- \frac{3 x^{8}}{2 {(5 - 3 x)}^{\frac{1}{2}}} + {(5 - 3 x)}^{\frac{1}{2}} \frac{d}{d x} [x^{8}]$

Step 16

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

$- \frac{3 x^{8}}{2 {(5 - 3 x)}^{\frac{1}{2}}} + {(5 - 3 x)}^{\frac{1}{2}} (8 x^{7})$

Step 17

Reorder.

Tap for more steps...

Step 17.1

Move $8$ to the left of ${(5 - 3 x)}^{\frac{1}{2}}$ .

$- \frac{3 x^{8}}{2 {(5 - 3 x)}^{\frac{1}{2}}} + 8 \cdot {(5 - 3 x)}^{\frac{1}{2}} x^{7}$

Step 17.2

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

$- \frac{3 x^{8}}{2 {(- 3 x + 5)}^{\frac{1}{2}}} + 8 x^{7} {(- 3 x + 5)}^{\frac{1}{2}}$

Step 18

To write $8 x^{7} {(- 3 x + 5)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{2 {(- 3 x + 5)}^{\frac{1}{2}}}{2 {(- 3 x + 5)}^{\frac{1}{2}}}$ .

$- \frac{3 x^{8}}{2 {(- 3 x + 5)}^{\frac{1}{2}}} + 8 x^{7} {(- 3 x + 5)}^{\frac{1}{2}} \cdot \frac{2 {(- 3 x + 5)}^{\frac{1}{2}}}{2 {(- 3 x + 5)}^{\frac{1}{2}}}$

Step 19

Combine $8 x^{7} {(- 3 x + 5)}^{\frac{1}{2}}$ and $\frac{2 {(- 3 x + 5)}^{\frac{1}{2}}}{2 {(- 3 x + 5)}^{\frac{1}{2}}}$ .

$- \frac{3 x^{8}}{2 {(- 3 x + 5)}^{\frac{1}{2}}} + \frac{8 x^{7} {(- 3 x + 5)}^{\frac{1}{2}} (2 {(- 3 x + 5)}^{\frac{1}{2}})}{2 {(- 3 x + 5)}^{\frac{1}{2}}}$

Step 20

Combine the numerators over the common denominator.

$\frac{- 3 x^{8} + 8 x^{7} {(- 3 x + 5)}^{\frac{1}{2}} (2 {(- 3 x + 5)}^{\frac{1}{2}})}{2 {(- 3 x + 5)}^{\frac{1}{2}}}$

Step 21

Multiply $2$ by $8$ .

$\frac{- 3 x^{8} + 16 x^{7} {(- 3 x + 5)}^{\frac{1}{2}} {(- 3 x + 5)}^{\frac{1}{2}}}{2 {(- 3 x + 5)}^{\frac{1}{2}}}$

Step 22

Multiply ${(- 3 x + 5)}^{\frac{1}{2}}$ by ${(- 3 x + 5)}^{\frac{1}{2}}$ by adding the exponents.

Tap for more steps...

Step 22.1

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

$\frac{- 3 x^{8} + 16 x^{7} ({(- 3 x + 5)}^{\frac{1}{2}} {(- 3 x + 5)}^{\frac{1}{2}})}{2 {(- 3 x + 5)}^{\frac{1}{2}}}$

Step 22.2

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

$\frac{- 3 x^{8} + 16 x^{7} {(- 3 x + 5)}^{\frac{1}{2} + \frac{1}{2}}}{2 {(- 3 x + 5)}^{\frac{1}{2}}}$

Step 22.3

Combine the numerators over the common denominator.

$\frac{- 3 x^{8} + 16 x^{7} {(- 3 x + 5)}^{\frac{1 + 1}{2}}}{2 {(- 3 x + 5)}^{\frac{1}{2}}}$

Step 22.4

Add $1$ and $1$ .

$\frac{- 3 x^{8} + 16 x^{7} {(- 3 x + 5)}^{\frac{2}{2}}}{2 {(- 3 x + 5)}^{\frac{1}{2}}}$

Step 22.5

Divide $2$ by $2$ .

$\frac{- 3 x^{8} + 16 x^{7} {(- 3 x + 5)}^{1}}{2 {(- 3 x + 5)}^{\frac{1}{2}}}$

Step 23

Simplify $16 x^{7} {(- 3 x + 5)}^{1}$ .

$\frac{- 3 x^{8} + 16 x^{7} (- 3 x + 5)}{2 {(- 3 x + 5)}^{\frac{1}{2}}}$

Step 24

Simplify.

Tap for more steps...

Step 24.1

Apply the distributive property.

$\frac{- 3 x^{8} + 16 x^{7} (- 3 x) + 16 x^{7} \cdot 5}{2 {(- 3 x + 5)}^{\frac{1}{2}}}$

Step 24.2

Simplify the numerator.

Tap for more steps...

Step 24.2.1

Simplify each term.

Tap for more steps...

Step 24.2.1.1

Rewrite using the commutative property of multiplication.

$\frac{- 3 x^{8} + 16 \cdot - 3 x^{7} x + 16 x^{7} \cdot 5}{2 {(- 3 x + 5)}^{\frac{1}{2}}}$

Step 24.2.1.2

Multiply $x^{7}$ by $x$ by adding the exponents.

Tap for more steps...

Step 24.2.1.2.1

Move $x$ .

$\frac{- 3 x^{8} + 16 \cdot - 3 (x \cdot x^{7}) + 16 x^{7} \cdot 5}{2 {(- 3 x + 5)}^{\frac{1}{2}}}$

Step 24.2.1.2.2

Multiply $x$ by $x^{7}$ .

Tap for more steps...

Step 24.2.1.2.2.1

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

$\frac{- 3 x^{8} + 16 \cdot - 3 (x^{1} x^{7}) + 16 x^{7} \cdot 5}{2 {(- 3 x + 5)}^{\frac{1}{2}}}$

Step 24.2.1.2.2.2

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

$\frac{- 3 x^{8} + 16 \cdot - 3 x^{1 + 7} + 16 x^{7} \cdot 5}{2 {(- 3 x + 5)}^{\frac{1}{2}}}$

Step 24.2.1.2.3

Add $1$ and $7$ .

$\frac{- 3 x^{8} + 16 \cdot - 3 x^{8} + 16 x^{7} \cdot 5}{2 {(- 3 x + 5)}^{\frac{1}{2}}}$

Step 24.2.1.3

Multiply $16$ by $- 3$ .

$\frac{- 3 x^{8} - 48 x^{8} + 16 x^{7} \cdot 5}{2 {(- 3 x + 5)}^{\frac{1}{2}}}$

Step 24.2.1.4

Multiply $5$ by $16$ .

$\frac{- 3 x^{8} - 48 x^{8} + 80 x^{7}}{2 {(- 3 x + 5)}^{\frac{1}{2}}}$

Step 24.2.2

Subtract $48 x^{8}$ from $- 3 x^{8}$ .

$\frac{- 51 x^{8} + 80 x^{7}}{2 {(- 3 x + 5)}^{\frac{1}{2}}}$

Step 24.3

Factor $x^{7}$ out of $- 51 x^{8} + 80 x^{7}$ .

Tap for more steps...

Step 24.3.1

Factor $x^{7}$ out of $- 51 x^{8}$ .

$\frac{x^{7} (- 51 x) + 80 x^{7}}{2 {(- 3 x + 5)}^{\frac{1}{2}}}$

Step 24.3.2

Factor $x^{7}$ out of $80 x^{7}$ .

$\frac{x^{7} (- 51 x) + x^{7} \cdot 80}{2 {(- 3 x + 5)}^{\frac{1}{2}}}$

Step 24.3.3

Factor $x^{7}$ out of $x^{7} (- 51 x) + x^{7} \cdot 80$ .

$\frac{x^{7} (- 51 x + 80)}{2 {(- 3 x + 5)}^{\frac{1}{2}}}$

Step 24.4

Factor $- 1$ out of $- 51 x$ .

$\frac{x^{7} (- (51 x) + 80)}{2 {(- 3 x + 5)}^{\frac{1}{2}}}$

Step 24.5

Rewrite $80$ as $- 1 (- 80)$ .

$\frac{x^{7} (- (51 x) - 1 (- 80))}{2 {(- 3 x + 5)}^{\frac{1}{2}}}$

Step 24.6

Factor $- 1$ out of $- (51 x) - 1 (- 80)$ .

$\frac{x^{7} (- (51 x - 80))}{2 {(- 3 x + 5)}^{\frac{1}{2}}}$

Step 24.7

Rewrite $- (51 x - 80)$ as $- 1 (51 x - 80)$ .

$\frac{x^{7} (- 1 (51 x - 80))}{2 {(- 3 x + 5)}^{\frac{1}{2}}}$

Step 24.8

Move the negative in front of the fraction.

$- \frac{x^{7} (51 x - 80)}{2 {(- 3 x + 5)}^{\frac{1}{2}}}$