Calculus Examples

$\frac{d y}{d x} = - \frac{x + 6}{7 y^{2}}$

Step 1

Separate the variables.

Tap for more steps...

Step 1.1

Multiply both sides by $y^{2}$ .

$y^{2} \frac{d y}{d x} = y^{2} (- \frac{x + 6}{7 y^{2}})$

Step 1.2

Simplify.

Tap for more steps...

Step 1.2.1

Rewrite using the commutative property of multiplication.

$y^{2} \frac{d y}{d x} = - y^{2} \frac{x + 6}{7 y^{2}}$

Step 1.2.2

Cancel the common factor of $y^{2}$ .

Tap for more steps...

Step 1.2.2.1

Factor $y^{2}$ out of $- y^{2}$ .

$y^{2} \frac{d y}{d x} = y^{2} \cdot - 1 \frac{x + 6}{7 y^{2}}$

Step 1.2.2.2

Factor $y^{2}$ out of $7 y^{2}$ .

$y^{2} \frac{d y}{d x} = y^{2} \cdot - 1 \frac{x + 6}{y^{2} \cdot 7}$

Step 1.2.2.3

Cancel the common factor.

$y^{2} \frac{d y}{d x} = y^{2} \cdot - 1 \frac{x + 6}{y^{2} \cdot 7}$

Step 1.2.2.4

Rewrite the expression.

$y^{2} \frac{d y}{d x} = - \frac{x + 6}{7}$

Step 1.3

Rewrite the equation.

$y^{2} d y = - \frac{x + 6}{7} d x$

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

$\int y^{2} d y = \int - \frac{x + 6}{7} d x$

Step 2.2

By the Power Rule, the integral of $y^{2}$ with respect to $y$ is $\frac{1}{3} y^{3}$ .

$\frac{1}{3} y^{3} + C_{1} = \int - \frac{x + 6}{7} d x$

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$\frac{1}{3} y^{3} + C_{1} = - \int \frac{x + 6}{7} d x$

Step 2.3.2

Since $\frac{1}{7}$ is constant with respect to $x$ , move $\frac{1}{7}$ out of the integral.

$\frac{1}{3} y^{3} + C_{1} = - (\frac{1}{7} \int x + 6 d x)$

Step 2.3.3

Split the single integral into multiple integrals.

$\frac{1}{3} y^{3} + C_{1} = - \frac{1}{7} (\int x d x + \int 6 d x)$

Step 2.3.4

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$\frac{1}{3} y^{3} + C_{1} = - \frac{1}{7} (\frac{1}{2} x^{2} + C_{2} + \int 6 d x)$

Step 2.3.5

Apply the constant rule.

$\frac{1}{3} y^{3} + C_{1} = - \frac{1}{7} (\frac{1}{2} x^{2} + C_{2} + 6 x + C_{3})$

Step 2.3.6

Simplify.

$\frac{1}{3} y^{3} + C_{1} = - \frac{1}{7} (\frac{1}{2} x^{2} + 6 x) + C_{4}$

Step 2.4

Group the constant of integration on the right side as $K$ .

$\frac{1}{3} y^{3} = - \frac{1}{7} (\frac{1}{2} x^{2} + 6 x) + K$

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Multiply both sides of the equation by $3$ .

$3 (\frac{1}{3} y^{3}) = 3 (- \frac{1}{7} (\frac{1}{2} x^{2} + 6 x) + K)$

Step 3.2

Simplify both sides of the equation.

Tap for more steps...

Step 3.2.1

Simplify the left side.

Tap for more steps...

Step 3.2.1.1

Simplify $3 (\frac{1}{3} y^{3})$ .

Tap for more steps...

Step 3.2.1.1.1

Combine $\frac{1}{3}$ and $y^{3}$ .

$3 \frac{y^{3}}{3} = 3 (- \frac{1}{7} (\frac{1}{2} x^{2} + 6 x) + K)$

Step 3.2.1.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.2.1.1.2.1

Cancel the common factor.

$3 \frac{y^{3}}{3} = 3 (- \frac{1}{7} (\frac{1}{2} x^{2} + 6 x) + K)$

Step 3.2.1.1.2.2

Rewrite the expression.

$y^{3} = 3 (- \frac{1}{7} (\frac{1}{2} x^{2} + 6 x) + K)$

Step 3.2.2

Simplify the right side.

Tap for more steps...

Step 3.2.2.1

Simplify $3 (- \frac{1}{7} (\frac{1}{2} x^{2} + 6 x) + K)$ .

Tap for more steps...

Step 3.2.2.1.1

Simplify each term.

Tap for more steps...

Step 3.2.2.1.1.1

Combine $\frac{1}{2}$ and $x^{2}$ .

$y^{3} = 3 (- \frac{1}{7} (\frac{x^{2}}{2} + 6 x) + K)$

Step 3.2.2.1.1.2

Apply the distributive property.

$y^{3} = 3 (- \frac{1}{7} \cdot \frac{x^{2}}{2} - \frac{1}{7} (6 x) + K)$

Step 3.2.2.1.1.3

Multiply $- \frac{1}{7} \cdot \frac{x^{2}}{2}$ .

Tap for more steps...

Step 3.2.2.1.1.3.1

Multiply $\frac{x^{2}}{2}$ by $\frac{1}{7}$ .

$y^{3} = 3 (- \frac{x^{2}}{2 \cdot 7} - \frac{1}{7} (6 x) + K)$

Step 3.2.2.1.1.3.2

Multiply $2$ by $7$ .

$y^{3} = 3 (- \frac{x^{2}}{14} - \frac{1}{7} (6 x) + K)$

Step 3.2.2.1.1.4

Multiply $- \frac{1}{7} (6 x)$ .

Tap for more steps...

Step 3.2.2.1.1.4.1

Multiply $6$ by $- 1$ .

$y^{3} = 3 (- \frac{x^{2}}{14} - 6 (\frac{1}{7}) x + K)$

Step 3.2.2.1.1.4.2

Combine $- 6$ and $\frac{1}{7}$ .

$y^{3} = 3 (- \frac{x^{2}}{14} + \frac{- 6}{7} x + K)$

Step 3.2.2.1.1.4.3

Combine $\frac{- 6}{7}$ and $x$ .

$y^{3} = 3 (- \frac{x^{2}}{14} + \frac{- 6 x}{7} + K)$

Step 3.2.2.1.1.5

Move the negative in front of the fraction.

$y^{3} = 3 (- \frac{x^{2}}{14} - \frac{6 x}{7} + K)$

Step 3.2.2.1.2

Apply the distributive property.

$y^{3} = 3 (- \frac{x^{2}}{14}) + 3 (- \frac{6 x}{7}) + 3 K$

Step 3.2.2.1.3

Simplify.

Tap for more steps...

Step 3.2.2.1.3.1

Multiply $3 (- \frac{x^{2}}{14})$ .

Tap for more steps...

Step 3.2.2.1.3.1.1

Multiply $- 1$ by $3$ .

$y^{3} = - 3 \frac{x^{2}}{14} + 3 (- \frac{6 x}{7}) + 3 K$

Step 3.2.2.1.3.1.2

Combine $- 3$ and $\frac{x^{2}}{14}$ .

$y^{3} = \frac{- 3 x^{2}}{14} + 3 (- \frac{6 x}{7}) + 3 K$

Step 3.2.2.1.3.2

Multiply $3 (- \frac{6 x}{7})$ .

Tap for more steps...

Step 3.2.2.1.3.2.1

Multiply $- 1$ by $3$ .

$y^{3} = \frac{- 3 x^{2}}{14} - 3 \frac{6 x}{7} + 3 K$

Step 3.2.2.1.3.2.2

Combine $- 3$ and $\frac{6 x}{7}$ .

$y^{3} = \frac{- 3 x^{2}}{14} + \frac{- 3 (6 x)}{7} + 3 K$

Step 3.2.2.1.3.2.3

Multiply $6$ by $- 3$ .

$y^{3} = \frac{- 3 x^{2}}{14} + \frac{- 18 x}{7} + 3 K$

Step 3.2.2.1.4

Simplify each term.

Tap for more steps...

Step 3.2.2.1.4.1

Move the negative in front of the fraction.

$y^{3} = - \frac{3 x^{2}}{14} + \frac{- 18 x}{7} + 3 K$

Step 3.2.2.1.4.2

Move the negative in front of the fraction.

$y^{3} = - \frac{3 x^{2}}{14} - \frac{18 x}{7} + 3 K$

Step 3.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \sqrt[3]{- \frac{3 x^{2}}{14} - \frac{18 x}{7} + 3 K}$

Step 3.4

Simplify $\sqrt[3]{- \frac{3 x^{2}}{14} - \frac{18 x}{7} + 3 K}$ .

Tap for more steps...

Step 3.4.1

To write $- \frac{18 x}{7}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \sqrt[3]{- \frac{3 x^{2}}{14} - \frac{18 x}{7} \cdot \frac{2}{2} + 3 K}$

Step 3.4.2

Write each expression with a common denominator of $14$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 3.4.2.1

Multiply $\frac{18 x}{7}$ by $\frac{2}{2}$ .

$y = \sqrt[3]{- \frac{3 x^{2}}{14} - \frac{18 x \cdot 2}{7 \cdot 2} + 3 K}$

Step 3.4.2.2

Multiply $7$ by $2$ .

$y = \sqrt[3]{- \frac{3 x^{2}}{14} - \frac{18 x \cdot 2}{14} + 3 K}$

Step 3.4.3

Combine the numerators over the common denominator.

$y = \sqrt[3]{\frac{- 3 x^{2} - 18 x \cdot 2}{14} + 3 K}$

Step 3.4.4

Simplify the numerator.

Tap for more steps...

Step 3.4.4.1

Factor $3 x$ out of $- 3 x^{2} - 18 x \cdot 2$ .

Tap for more steps...

Step 3.4.4.1.1

Factor $3 x$ out of $- 3 x^{2}$ .

$y = \sqrt[3]{\frac{3 x (- x) - 18 x \cdot 2}{14} + 3 K}$

Step 3.4.4.1.2

Factor $3 x$ out of $- 18 x \cdot 2$ .

$y = \sqrt[3]{\frac{3 x (- x) + 3 x (- 6 \cdot 2)}{14} + 3 K}$

Step 3.4.4.1.3

Factor $3 x$ out of $3 x (- x) + 3 x (- 6 \cdot 2)$ .

$y = \sqrt[3]{\frac{3 x (- x - 6 \cdot 2)}{14} + 3 K}$

Step 3.4.4.2

Multiply $- 6$ by $2$ .

$y = \sqrt[3]{\frac{3 x (- x - 12)}{14} + 3 K}$

Step 3.4.5

To write $3 K$ as a fraction with a common denominator, multiply by $\frac{14}{14}$ .

$y = \sqrt[3]{\frac{3 x (- x - 12)}{14} + 3 K \cdot \frac{14}{14}}$

Step 3.4.6

Simplify terms.

Tap for more steps...

Step 3.4.6.1

Combine $3 K$ and $\frac{14}{14}$ .

$y = \sqrt[3]{\frac{3 x (- x - 12)}{14} + \frac{3 K \cdot 14}{14}}$

Step 3.4.6.2

Combine the numerators over the common denominator.

$y = \sqrt[3]{\frac{3 x (- x - 12) + 3 K \cdot 14}{14}}$

Step 3.4.7

Simplify the numerator.

Tap for more steps...

Step 3.4.7.1

Factor $3$ out of $3 x (- x - 12) + 3 K \cdot 14$ .

Tap for more steps...

Step 3.4.7.1.1

Factor $3$ out of $3 x (- x - 12)$ .

$y = \sqrt[3]{\frac{3 (x (- x - 12)) + 3 K \cdot 14}{14}}$

Step 3.4.7.1.2

Factor $3$ out of $3 K \cdot 14$ .

$y = \sqrt[3]{\frac{3 (x (- x - 12)) + 3 (K \cdot 14)}{14}}$

Step 3.4.7.1.3

Factor $3$ out of $3 (x (- x - 12)) + 3 (K \cdot 14)$ .

$y = \sqrt[3]{\frac{3 (x (- x - 12) + K \cdot 14)}{14}}$

Step 3.4.7.2

Apply the distributive property.

$y = \sqrt[3]{\frac{3 (x (- x) + x \cdot - 12 + K \cdot 14)}{14}}$

Step 3.4.7.3

Rewrite using the commutative property of multiplication.

$y = \sqrt[3]{\frac{3 (- x \cdot x + x \cdot - 12 + K \cdot 14)}{14}}$

Step 3.4.7.4

Move $- 12$ to the left of $x$ .

$y = \sqrt[3]{\frac{3 (- x \cdot x - 12 \cdot x + K \cdot 14)}{14}}$

Step 3.4.7.5

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 3.4.7.5.1

Move $x$ .

$y = \sqrt[3]{\frac{3 (- (x \cdot x) - 12 \cdot x + K \cdot 14)}{14}}$

Step 3.4.7.5.2

Multiply $x$ by $x$ .

$y = \sqrt[3]{\frac{3 (- x^{2} - 12 \cdot x + K \cdot 14)}{14}}$

$y = \sqrt[3]{\frac{3 (- x^{2} - 12 x + K \cdot 14)}{14}}$

Step 3.4.7.6

Move $14$ to the left of $K$ .

$y = \sqrt[3]{\frac{3 (- x^{2} - 12 x + 14 K)}{14}}$

Step 3.4.8

Rewrite $\sqrt[3]{\frac{3 (- x^{2} - 12 x + 14 K)}{14}}$ as $\frac{\sqrt[3]{3 (- x^{2} - 12 x + 14 K)}}{\sqrt[3]{14}}$ .

$y = \frac{\sqrt[3]{3 (- x^{2} - 12 x + 14 K)}}{\sqrt[3]{14}}$

Step 3.4.9

Multiply $\frac{\sqrt[3]{3 (- x^{2} - 12 x + 14 K)}}{\sqrt[3]{14}}$ by $\frac{{\sqrt[3]{14}}^{2}}{{\sqrt[3]{14}}^{2}}$ .

$y = \frac{\sqrt[3]{3 (- x^{2} - 12 x + 14 K)}}{\sqrt[3]{14}} \cdot \frac{{\sqrt[3]{14}}^{2}}{{\sqrt[3]{14}}^{2}}$

Step 3.4.10

Combine and simplify the denominator.

Tap for more steps...

Step 3.4.10.1

Multiply $\frac{\sqrt[3]{3 (- x^{2} - 12 x + 14 K)}}{\sqrt[3]{14}}$ by $\frac{{\sqrt[3]{14}}^{2}}{{\sqrt[3]{14}}^{2}}$ .

$y = \frac{\sqrt[3]{3 (- x^{2} - 12 x + 14 K)} {\sqrt[3]{14}}^{2}}{\sqrt[3]{14} {\sqrt[3]{14}}^{2}}$

Step 3.4.10.2

Raise $\sqrt[3]{14}$ to the power of $1$ .

$y = \frac{\sqrt[3]{3 (- x^{2} - 12 x + 14 K)} {\sqrt[3]{14}}^{2}}{{\sqrt[3]{14}}^{1} {\sqrt[3]{14}}^{2}}$

Step 3.4.10.3

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

$y = \frac{\sqrt[3]{3 (- x^{2} - 12 x + 14 K)} {\sqrt[3]{14}}^{2}}{{\sqrt[3]{14}}^{1 + 2}}$

Step 3.4.10.4

Add $1$ and $2$ .

$y = \frac{\sqrt[3]{3 (- x^{2} - 12 x + 14 K)} {\sqrt[3]{14}}^{2}}{{\sqrt[3]{14}}^{3}}$

Step 3.4.10.5

Rewrite ${\sqrt[3]{14}}^{3}$ as $14$ .

Tap for more steps...

Step 3.4.10.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{14}$ as $14^{\frac{1}{3}}$ .

$y = \frac{\sqrt[3]{3 (- x^{2} - 12 x + 14 K)} {\sqrt[3]{14}}^{2}}{{(14^{\frac{1}{3}})}^{3}}$

Step 3.4.10.5.2

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

$y = \frac{\sqrt[3]{3 (- x^{2} - 12 x + 14 K)} {\sqrt[3]{14}}^{2}}{14^{\frac{1}{3} \cdot 3}}$

Step 3.4.10.5.3

Combine $\frac{1}{3}$ and $3$ .

$y = \frac{\sqrt[3]{3 (- x^{2} - 12 x + 14 K)} {\sqrt[3]{14}}^{2}}{14^{\frac{3}{3}}}$

Step 3.4.10.5.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.4.10.5.4.1

Cancel the common factor.

$y = \frac{\sqrt[3]{3 (- x^{2} - 12 x + 14 K)} {\sqrt[3]{14}}^{2}}{14^{\frac{3}{3}}}$

Step 3.4.10.5.4.2

Rewrite the expression.

$y = \frac{\sqrt[3]{3 (- x^{2} - 12 x + 14 K)} {\sqrt[3]{14}}^{2}}{14^{1}}$

Step 3.4.10.5.5

Evaluate the exponent.

$y = \frac{\sqrt[3]{3 (- x^{2} - 12 x + 14 K)} {\sqrt[3]{14}}^{2}}{14}$

Step 3.4.11

Simplify the numerator.

Tap for more steps...

Step 3.4.11.1

Rewrite ${\sqrt[3]{14}}^{2}$ as $\sqrt[3]{14^{2}}$ .

$y = \frac{\sqrt[3]{3 (- x^{2} - 12 x + 14 K)} \sqrt[3]{14^{2}}}{14}$

Step 3.4.11.2

Raise $14$ to the power of $2$ .

$y = \frac{\sqrt[3]{3 (- x^{2} - 12 x + 14 K)} \sqrt[3]{196}}{14}$

Step 3.4.12

Simplify the numerator.

Tap for more steps...

Step 3.4.12.1

Combine using the product rule for radicals.

$y = \frac{\sqrt[3]{3 (- x^{2} - 12 x + 14 K) \cdot 196}}{14}$

Step 3.4.12.2

Multiply $196$ by $3$ .

$y = \frac{\sqrt[3]{588 (- x^{2} - 12 x + 14 K)}}{14}$

Step 4

Simplify the constant of integration.

$y = \frac{\sqrt[3]{588 (- x^{2} - 12 x + K)}}{14}$