Calculus Examples

$q = \sqrt{15 r - r^{7}}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15 r - r^{7}}$ as ${(15 r - r^{7})}^{\frac{1}{2}}$ .

$q = {(15 r - r^{7})}^{\frac{1}{2}}$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d q} (q) = \frac{d}{d q} ({(15 r - r^{7})}^{\frac{1}{2}})$

Step 3

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

$1$

Step 4

Differentiate the right side of the equation.

Tap for more steps...

Step 4.1

Differentiate using the chain rule, which states that $\frac{d}{d q} [f (g (q))]$ is $f' (g (q)) g' (q)$ where $f (q) = q^{\frac{1}{2}}$ and $g (q) = 15 r - r^{7}$ .

Tap for more steps...

Step 4.1.1

To apply the Chain Rule, set $u_{1}$ as $15 r - r^{7}$ .

$\frac{d}{d u_{1}} [{u_{1}}^{\frac{1}{2}}] \frac{d}{d q} [15 r - r^{7}]$

Step 4.1.2

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

$\frac{1}{2} {u_{1}}^{\frac{1}{2} - 1} \frac{d}{d q} [15 r - r^{7}]$

Step 4.1.3

Replace all occurrences of $u_{1}$ with $15 r - r^{7}$ .

$\frac{1}{2} {(15 r - r^{7})}^{\frac{1}{2} - 1} \frac{d}{d q} [15 r - r^{7}]$

Step 4.2

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

$\frac{1}{2} {(15 r - r^{7})}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d q} [15 r - r^{7}]$

Step 4.3

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

$\frac{1}{2} {(15 r - r^{7})}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d q} [15 r - r^{7}]$

Step 4.4

Combine the numerators over the common denominator.

$\frac{1}{2} {(15 r - r^{7})}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d q} [15 r - r^{7}]$

Step 4.5

Simplify the numerator.

Tap for more steps...

Step 4.5.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} {(15 r - r^{7})}^{\frac{1 - 2}{2}} \frac{d}{d q} [15 r - r^{7}]$

Step 4.5.2

Subtract $2$ from $1$ .

$\frac{1}{2} {(15 r - r^{7})}^{\frac{- 1}{2}} \frac{d}{d q} [15 r - r^{7}]$

Step 4.6

Combine fractions.

Tap for more steps...

Step 4.6.1

Move the negative in front of the fraction.

$\frac{1}{2} {(15 r - r^{7})}^{- \frac{1}{2}} \frac{d}{d q} [15 r - r^{7}]$

Step 4.6.2

Combine $\frac{1}{2}$ and ${(15 r - r^{7})}^{- \frac{1}{2}}$ .

$\frac{{(15 r - r^{7})}^{- \frac{1}{2}}}{2} \frac{d}{d q} [15 r - r^{7}]$

Step 4.6.3

Move ${(15 r - r^{7})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2 {(15 r - r^{7})}^{\frac{1}{2}}} \frac{d}{d q} [15 r - r^{7}]$

Step 4.7

By the Sum Rule, the derivative of $15 r - r^{7}$ with respect to $q$ is $\frac{d}{d q} [15 r] + \frac{d}{d q} [- r^{7}]$ .

$\frac{1}{2 {(15 r - r^{7})}^{\frac{1}{2}}} (\frac{d}{d q} [15 r] + \frac{d}{d q} [- r^{7}])$

Step 4.8

Since $15$ is constant with respect to $q$ , the derivative of $15 r$ with respect to $q$ is $15 \frac{d}{d q} [r]$ .

$\frac{1}{2 {(15 r - r^{7})}^{\frac{1}{2}}} (15 \frac{d}{d q} [r] + \frac{d}{d q} [- r^{7}])$

Step 4.9

Rewrite $\frac{d}{d q} [r]$ as $r'$ .

$\frac{1}{2 {(15 r - r^{7})}^{\frac{1}{2}}} (15 r' + \frac{d}{d q} [- r^{7}])$

Step 4.10

Since $- 1$ is constant with respect to $q$ , the derivative of $- r^{7}$ with respect to $q$ is $- \frac{d}{d q} [r^{7}]$ .

$\frac{1}{2 {(15 r - r^{7})}^{\frac{1}{2}}} (15 r' - \frac{d}{d q} [r^{7}])$

Step 4.11

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

Tap for more steps...

Step 4.11.1

To apply the Chain Rule, set $u_{2}$ as $r$ .

$\frac{1}{2 {(15 r - r^{7})}^{\frac{1}{2}}} (15 r' - (\frac{d}{d u_{2}} [{u_{2}}^{7}] \frac{d}{d q} [r]))$

Step 4.11.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 7$ .

$\frac{1}{2 {(15 r - r^{7})}^{\frac{1}{2}}} (15 r' - (7 {u_{2}}^{6} \frac{d}{d q} [r]))$

Step 4.11.3

Replace all occurrences of $u_{2}$ with $r$ .

$\frac{1}{2 {(15 r - r^{7})}^{\frac{1}{2}}} (15 r' - (7 r^{6} \frac{d}{d q} [r]))$

Step 4.12

Multiply $7$ by $- 1$ .

$\frac{1}{2 {(15 r - r^{7})}^{\frac{1}{2}}} (15 r' - 7 (r^{6} \frac{d}{d q} [r]))$

Step 4.13

Rewrite $\frac{d}{d q} [r]$ as $r'$ .

$\frac{1}{2 {(15 r - r^{7})}^{\frac{1}{2}}} (15 r' - 7 r^{6} r')$

Step 4.14

Simplify.

Tap for more steps...

Step 4.14.1

Apply the distributive property.

$\frac{1}{2 {(15 r - r^{7})}^{\frac{1}{2}}} (15 r') + \frac{1}{2 {(15 r - r^{7})}^{\frac{1}{2}}} (- 7 r^{6} r')$

Step 4.14.2

Combine terms.

Tap for more steps...

Step 4.14.2.1

Combine $15$ and $\frac{1}{2 {(15 r - r^{7})}^{\frac{1}{2}}}$ .

$\frac{15}{2 {(15 r - r^{7})}^{\frac{1}{2}}} r' + \frac{1}{2 {(15 r - r^{7})}^{\frac{1}{2}}} (- 7 r^{6} r')$

Step 4.14.2.2

Combine $\frac{15}{2 {(15 r - r^{7})}^{\frac{1}{2}}}$ and $r'$ .

$\frac{15 r'}{2 {(15 r - r^{7})}^{\frac{1}{2}}} + \frac{1}{2 {(15 r - r^{7})}^{\frac{1}{2}}} (- 7 r^{6} r')$

Step 4.14.2.3

Combine $- 7$ and $\frac{1}{2 {(15 r - r^{7})}^{\frac{1}{2}}}$ .

$\frac{15 r'}{2 {(15 r - r^{7})}^{\frac{1}{2}}} + \frac{- 7}{2 {(15 r - r^{7})}^{\frac{1}{2}}} (r^{6} r')$

Step 4.14.2.4

Combine $r^{6}$ and $\frac{- 7}{2 {(15 r - r^{7})}^{\frac{1}{2}}}$ .

$\frac{15 r'}{2 {(15 r - r^{7})}^{\frac{1}{2}}} + \frac{r^{6} \cdot - 7}{2 {(15 r - r^{7})}^{\frac{1}{2}}} r'$

Step 4.14.2.5

Combine $\frac{r^{6} \cdot - 7}{2 {(15 r - r^{7})}^{\frac{1}{2}}}$ and $r'$ .

$\frac{15 r'}{2 {(15 r - r^{7})}^{\frac{1}{2}}} + \frac{r^{6} \cdot - 7 r'}{2 {(15 r - r^{7})}^{\frac{1}{2}}}$

Step 4.14.2.6

Move $- 7$ to the left of $r^{6}$ .

$\frac{15 r'}{2 {(15 r - r^{7})}^{\frac{1}{2}}} + \frac{- 7 \cdot r^{6} r'}{2 {(15 r - r^{7})}^{\frac{1}{2}}}$

Step 4.14.2.7

Move the negative in front of the fraction.

$\frac{15 r'}{2 {(15 r - r^{7})}^{\frac{1}{2}}} - \frac{7 r^{6} r'}{2 {(15 r - r^{7})}^{\frac{1}{2}}}$

Step 4.14.3

Reorder terms.

$- \frac{7 r^{6} r'}{2 {(15 r - r^{7})}^{\frac{1}{2}}} + \frac{15 r'}{2 {(15 r - r^{7})}^{\frac{1}{2}}}$

Step 5

Reform the equation by setting the left side equal to the right side.

$1 = - \frac{7 r^{6} r'}{2 {(15 r - r^{7})}^{\frac{1}{2}}} + \frac{15 r'}{2 {(15 r - r^{7})}^{\frac{1}{2}}}$

Step 6

Solve for $r'$ .

Tap for more steps...

Step 6.1

Rewrite the equation as $- \frac{7 r^{6} r'}{2 {(15 r - r^{7})}^{\frac{1}{2}}} + \frac{15 r'}{2 {(15 r - r^{7})}^{\frac{1}{2}}} = 1$ .

$- \frac{7 r^{6} r'}{2 {(15 r - r^{7})}^{\frac{1}{2}}} + \frac{15 r'}{2 {(15 r - r^{7})}^{\frac{1}{2}}} = 1$

Step 6.2

Find the LCD of the terms in the equation.

Tap for more steps...

Step 6.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$2 {(15 r - r^{7})}^{\frac{1}{2}}, 2 {(15 r - r^{7})}^{\frac{1}{2}}, 1$

Step 6.2.2

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 6.2.3

Since $2$ has no factors besides $1$ and $2$ .

$2$ is a prime number

Step 6.2.4

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 6.2.5

The LCM of $2, 2, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$2$

Step 6.2.6

The factor for $15 r - r^{7}$ is $15 r - r^{7}$ itself.

$(15 r - r^{7}) = 15 r - r^{7}$

$(15 r - r^{7})$ occurs $1$ time.

Step 6.2.7

The LCM of ${(15 r - r^{7})}^{\frac{1}{2}}, {(15 r - r^{7})}^{\frac{1}{2}}$ is the result of multiplying all factors the greatest number of times they occur in either term.

${(15 r - r^{7})}^{\frac{1}{2}}$

Step 6.2.8

The Least Common Multiple $LCM$ of some numbers is the smallest number that the numbers are factors of.

$2 {(15 r - r^{7})}^{\frac{1}{2}}$

Step 6.3

Multiply each term in $- \frac{7 r^{6} r'}{2 {(15 r - r^{7})}^{\frac{1}{2}}} + \frac{15 r'}{2 {(15 r - r^{7})}^{\frac{1}{2}}} = 1$ by $2 {(15 r - r^{7})}^{\frac{1}{2}}$ to eliminate the fractions.

Tap for more steps...

Step 6.3.1

Multiply each term in $- \frac{7 r^{6} r'}{2 {(15 r - r^{7})}^{\frac{1}{2}}} + \frac{15 r'}{2 {(15 r - r^{7})}^{\frac{1}{2}}} = 1$ by $2 {(15 r - r^{7})}^{\frac{1}{2}}$ .

$- \frac{7 r^{6} r'}{2 {(15 r - r^{7})}^{\frac{1}{2}}} (2 {(15 r - r^{7})}^{\frac{1}{2}}) + \frac{15 r'}{2 {(15 r - r^{7})}^{\frac{1}{2}}} (2 {(15 r - r^{7})}^{\frac{1}{2}}) = 1 (2 {(15 r - r^{7})}^{\frac{1}{2}})$

Step 6.3.2

Simplify the left side.

Tap for more steps...

Step 6.3.2.1

Simplify each term.

Tap for more steps...

Step 6.3.2.1.1

Cancel the common factor of $2 {(15 r - r^{7})}^{\frac{1}{2}}$ .

Tap for more steps...

Step 6.3.2.1.1.1

Move the leading negative in $- \frac{7 r^{6} r'}{2 {(15 r - r^{7})}^{\frac{1}{2}}}$ into the numerator.

$\frac{- 7 r^{6} r'}{2 {(15 r - r^{7})}^{\frac{1}{2}}} (2 {(15 r - r^{7})}^{\frac{1}{2}}) + \frac{15 r'}{2 {(15 r - r^{7})}^{\frac{1}{2}}} (2 {(15 r - r^{7})}^{\frac{1}{2}}) = 1 (2 {(15 r - r^{7})}^{\frac{1}{2}})$

Step 6.3.2.1.1.2

Cancel the common factor.

Step 6.3.2.1.1.3

Rewrite the expression.

$- 7 r^{6} r' + \frac{15 r'}{2 {(15 r - r^{7})}^{\frac{1}{2}}} (2 {(15 r - r^{7})}^{\frac{1}{2}}) = 1 (2 {(15 r - r^{7})}^{\frac{1}{2}})$

Step 6.3.2.1.2

Rewrite using the commutative property of multiplication.

$- 7 r^{6} r' + 2 \frac{15 r'}{2 {(15 r - r^{7})}^{\frac{1}{2}}} {(15 r - r^{7})}^{\frac{1}{2}} = 1 (2 {(15 r - r^{7})}^{\frac{1}{2}})$

Step 6.3.2.1.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.3.2.1.3.1

Cancel the common factor.

$- 7 r^{6} r' + 2 \frac{15 r'}{2 {(15 r - r^{7})}^{\frac{1}{2}}} {(15 r - r^{7})}^{\frac{1}{2}} = 1 (2 {(15 r - r^{7})}^{\frac{1}{2}})$

Step 6.3.2.1.3.2

Rewrite the expression.

$- 7 r^{6} r' + \frac{15 r'}{{(15 r - r^{7})}^{\frac{1}{2}}} {(15 r - r^{7})}^{\frac{1}{2}} = 1 (2 {(15 r - r^{7})}^{\frac{1}{2}})$

Step 6.3.2.1.4

Cancel the common factor of ${(15 r - r^{7})}^{\frac{1}{2}}$ .

Tap for more steps...

Step 6.3.2.1.4.1

Cancel the common factor.

$- 7 r^{6} r' + \frac{15 r'}{{(15 r - r^{7})}^{\frac{1}{2}}} {(15 r - r^{7})}^{\frac{1}{2}} = 1 (2 {(15 r - r^{7})}^{\frac{1}{2}})$

Step 6.3.2.1.4.2

Rewrite the expression.

$- 7 r^{6} r' + 15 r' = 1 (2 {(15 r - r^{7})}^{\frac{1}{2}})$

Step 6.3.3

Simplify the right side.

Tap for more steps...

Step 6.3.3.1

Multiply $2 {(15 r - r^{7})}^{\frac{1}{2}}$ by $1$ .

$- 7 r^{6} r' + 15 r' = 2 {(15 r - r^{7})}^{\frac{1}{2}}$

Step 6.4

Solve the equation.

Tap for more steps...

Step 6.4.1

Factor $r'$ out of $- 7 r^{6} r' + 15 r'$ .

Tap for more steps...

Step 6.4.1.1

Factor $r'$ out of $- 7 r^{6} r'$ .

$r' (- 7 r^{6}) + 15 r' = 2 {(15 r - r^{7})}^{\frac{1}{2}}$

Step 6.4.1.2

Factor $r'$ out of $15 r'$ .

$r' (- 7 r^{6}) + r' \cdot 15 = 2 {(15 r - r^{7})}^{\frac{1}{2}}$

Step 6.4.1.3

Factor $r'$ out of $r' (- 7 r^{6}) + r' \cdot 15$ .

$r' (- 7 r^{6} + 15) = 2 {(15 r - r^{7})}^{\frac{1}{2}}$

Step 6.4.2

Divide each term in $r' (- 7 r^{6} + 15) = 2 {(15 r - r^{7})}^{\frac{1}{2}}$ by $- 7 r^{6} + 15$ and simplify.

Tap for more steps...

Step 6.4.2.1

Divide each term in $r' (- 7 r^{6} + 15) = 2 {(15 r - r^{7})}^{\frac{1}{2}}$ by $- 7 r^{6} + 15$ .

$\frac{r' (- 7 r^{6} + 15)}{- 7 r^{6} + 15} = \frac{2 {(15 r - r^{7})}^{\frac{1}{2}}}{- 7 r^{6} + 15}$

Step 6.4.2.2

Simplify the left side.

Tap for more steps...

Step 6.4.2.2.1

Cancel the common factor of $- 7 r^{6} + 15$ .

Tap for more steps...

Step 6.4.2.2.1.1

Cancel the common factor.

$\frac{r' (- 7 r^{6} + 15)}{- 7 r^{6} + 15} = \frac{2 {(15 r - r^{7})}^{\frac{1}{2}}}{- 7 r^{6} + 15}$

Step 6.4.2.2.1.2

Divide $r'$ by $1$ .

$r' = \frac{2 {(15 r - r^{7})}^{\frac{1}{2}}}{- 7 r^{6} + 15}$

Step 6.4.2.3

Simplify the right side.

Tap for more steps...

Step 6.4.2.3.1

Factor $- 1$ out of $- 7 r^{6}$ .

$r' = \frac{2 {(15 r - r^{7})}^{\frac{1}{2}}}{- (7 r^{6}) + 15}$

Step 6.4.2.3.2

Rewrite $15$ as $- 1 (- 15)$ .

$r' = \frac{2 {(15 r - r^{7})}^{\frac{1}{2}}}{- (7 r^{6}) - 1 (- 15)}$

Step 6.4.2.3.3

Factor $- 1$ out of $- (7 r^{6}) - 1 (- 15)$ .

$r' = \frac{2 {(15 r - r^{7})}^{\frac{1}{2}}}{- (7 r^{6} - 15)}$

Step 6.4.2.3.4

Rewrite negatives.

Tap for more steps...

Step 6.4.2.3.4.1

Rewrite $- (7 r^{6} - 15)$ as $- 1 (7 r^{6} - 15)$ .

$r' = \frac{2 {(15 r - r^{7})}^{\frac{1}{2}}}{- 1 (7 r^{6} - 15)}$

Step 6.4.2.3.4.2

Move the negative in front of the fraction.

$r' = - \frac{2 {(15 r - r^{7})}^{\frac{1}{2}}}{7 r^{6} - 15}$

Step 7

Replace $r'$ with $\frac{d r}{d q}$ .

$\frac{d r}{d q} = - \frac{2 {(15 r - r^{7})}^{\frac{1}{2}}}{7 r^{6} - 15}$