Calculus Examples

$\frac{x}{51 y^{2}} = \frac{d y}{d x}$

Step 1

Separate the variables.

Tap for more steps...

Step 1.1

Flip sides to get $\frac{d y}{d x}$ on the left side.

$\frac{d y}{d x} = \frac{x}{51 y^{2}}$

Step 1.2

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

$y^{2} \frac{d y}{d x} = y^{2} \frac{x}{51 y^{2}}$

Step 1.3

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

Tap for more steps...

Step 1.3.1

Factor $y^{2}$ out of $51 y^{2}$ .

$y^{2} \frac{d y}{d x} = y^{2} \frac{x}{y^{2} \cdot 51}$

Step 1.3.2

Cancel the common factor.

$y^{2} \frac{d y}{d x} = y^{2} \frac{x}{y^{2} \cdot 51}$

Step 1.3.3

Rewrite the expression.

$y^{2} \frac{d y}{d x} = \frac{x}{51}$

Step 1.4

Rewrite the equation.

$y^{2} d y = \frac{x}{51} d x$

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

$\int y^{2} d y = \int \frac{x}{51} d x$

Step 2.2

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

$\frac{1}{3} y^{3} + C_{1} = \int \frac{x}{51} d x$

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

Since $\frac{1}{51}$ is constant with respect to $x$ , move $\frac{1}{51}$ out of the integral.

$\frac{1}{3} y^{3} + C_{1} = \frac{1}{51} \int x d x$

Step 2.3.2

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

$\frac{1}{3} y^{3} + C_{1} = \frac{1}{51} (\frac{1}{2} x^{2} + C_{2})$

Step 2.3.3

Simplify the answer.

Tap for more steps...

Step 2.3.3.1

Rewrite $\frac{1}{51} (\frac{1}{2} x^{2} + C_{2})$ as $\frac{1}{51} \cdot \frac{1}{2} x^{2} + C_{2}$ .

$\frac{1}{3} y^{3} + C_{1} = \frac{1}{51} \cdot \frac{1}{2} x^{2} + C_{2}$

Step 2.3.3.2

Simplify.

Tap for more steps...

Step 2.3.3.2.1

Multiply $\frac{1}{51}$ by $\frac{1}{2}$ .

$\frac{1}{3} y^{3} + C_{1} = \frac{1}{51 \cdot 2} x^{2} + C_{2}$

Step 2.3.3.2.2

Multiply $51$ by $2$ .

$\frac{1}{3} y^{3} + C_{1} = \frac{1}{102} x^{2} + C_{2}$

Step 2.4

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

$\frac{1}{3} y^{3} = \frac{1}{102} x^{2} + K$

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Multiply both sides of the equation by $3$ .

$3 (\frac{1}{3} y^{3}) = 3 (\frac{1}{102} x^{2} + K)$

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 $y^{3}$ .

$3 \frac{y^{3}}{3} = 3 (\frac{1}{102} x^{2} + K)$

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.

$3 \frac{y^{3}}{3} = 3 (\frac{1}{102} x^{2} + K)$

Step 3.2.1.1.2.2

Rewrite the expression.

$y^{3} = 3 (\frac{1}{102} x^{2} + K)$

Step 3.2.2

Simplify the right side.

Tap for more steps...

Step 3.2.2.1

Simplify $3 (\frac{1}{102} x^{2} + K)$ .

Tap for more steps...

Step 3.2.2.1.1

Combine $\frac{1}{102}$ and $x^{2}$ .

$y^{3} = 3 (\frac{x^{2}}{102} + K)$

Step 3.2.2.1.2

Apply the distributive property.

$y^{3} = 3 \frac{x^{2}}{102} + 3 K$

Step 3.2.2.1.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.2.2.1.3.1

Factor $3$ out of $102$ .

$y^{3} = 3 \frac{x^{2}}{3 (34)} + 3 K$

Step 3.2.2.1.3.2

Cancel the common factor.

$y^{3} = 3 \frac{x^{2}}{3 \cdot 34} + 3 K$

Step 3.2.2.1.3.3

Rewrite the expression.

$y^{3} = \frac{x^{2}}{34} + 3 K$

Step 3.3

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

$y = \sqrt[3]{\frac{x^{2}}{34} + 3 K}$

Step 3.4

Simplify $\sqrt[3]{\frac{x^{2}}{34} + 3 K}$ .

Tap for more steps...

Step 3.4.1

To write $3 K$ as a fraction with a common denominator, multiply by $\frac{34}{34}$ .

$y = \sqrt[3]{\frac{x^{2}}{34} + 3 K \cdot \frac{34}{34}}$

Step 3.4.2

Combine $3 K$ and $\frac{34}{34}$ .

$y = \sqrt[3]{\frac{x^{2}}{34} + \frac{3 K \cdot 34}{34}}$

Step 3.4.3

Combine the numerators over the common denominator.

$y = \sqrt[3]{\frac{x^{2} + 3 K \cdot 34}{34}}$

Step 3.4.4

Multiply $34$ by $3$ .

$y = \sqrt[3]{\frac{x^{2} + 102 K}{34}}$

Step 3.4.5

Rewrite $\sqrt[3]{\frac{x^{2} + 102 K}{34}}$ as $\frac{\sqrt[3]{x^{2} + 102 K}}{\sqrt[3]{34}}$ .

$y = \frac{\sqrt[3]{x^{2} + 102 K}}{\sqrt[3]{34}}$

Step 3.4.6

Multiply $\frac{\sqrt[3]{x^{2} + 102 K}}{\sqrt[3]{34}}$ by $\frac{{\sqrt[3]{34}}^{2}}{{\sqrt[3]{34}}^{2}}$ .

$y = \frac{\sqrt[3]{x^{2} + 102 K}}{\sqrt[3]{34}} \cdot \frac{{\sqrt[3]{34}}^{2}}{{\sqrt[3]{34}}^{2}}$

Step 3.4.7

Combine and simplify the denominator.

Tap for more steps...

Step 3.4.7.1

Multiply $\frac{\sqrt[3]{x^{2} + 102 K}}{\sqrt[3]{34}}$ by $\frac{{\sqrt[3]{34}}^{2}}{{\sqrt[3]{34}}^{2}}$ .

$y = \frac{\sqrt[3]{x^{2} + 102 K} {\sqrt[3]{34}}^{2}}{\sqrt[3]{34} {\sqrt[3]{34}}^{2}}$

Step 3.4.7.2

Raise $\sqrt[3]{34}$ to the power of $1$ .

$y = \frac{\sqrt[3]{x^{2} + 102 K} {\sqrt[3]{34}}^{2}}{{\sqrt[3]{34}}^{1} {\sqrt[3]{34}}^{2}}$

Step 3.4.7.3

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

$y = \frac{\sqrt[3]{x^{2} + 102 K} {\sqrt[3]{34}}^{2}}{{\sqrt[3]{34}}^{1 + 2}}$

Step 3.4.7.4

Add $1$ and $2$ .

$y = \frac{\sqrt[3]{x^{2} + 102 K} {\sqrt[3]{34}}^{2}}{{\sqrt[3]{34}}^{3}}$

Step 3.4.7.5

Rewrite ${\sqrt[3]{34}}^{3}$ as $34$ .

Tap for more steps...

Step 3.4.7.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{34}$ as $34^{\frac{1}{3}}$ .

$y = \frac{\sqrt[3]{x^{2} + 102 K} {\sqrt[3]{34}}^{2}}{{(34^{\frac{1}{3}})}^{3}}$

Step 3.4.7.5.2

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

$y = \frac{\sqrt[3]{x^{2} + 102 K} {\sqrt[3]{34}}^{2}}{34^{\frac{1}{3} \cdot 3}}$

Step 3.4.7.5.3

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

$y = \frac{\sqrt[3]{x^{2} + 102 K} {\sqrt[3]{34}}^{2}}{34^{\frac{3}{3}}}$

Step 3.4.7.5.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.4.7.5.4.1

Cancel the common factor.

$y = \frac{\sqrt[3]{x^{2} + 102 K} {\sqrt[3]{34}}^{2}}{34^{\frac{3}{3}}}$

Step 3.4.7.5.4.2

Rewrite the expression.

$y = \frac{\sqrt[3]{x^{2} + 102 K} {\sqrt[3]{34}}^{2}}{34^{1}}$

Step 3.4.7.5.5

Evaluate the exponent.

$y = \frac{\sqrt[3]{x^{2} + 102 K} {\sqrt[3]{34}}^{2}}{34}$

Step 3.4.8

Simplify the numerator.

Tap for more steps...

Step 3.4.8.1

Rewrite ${\sqrt[3]{34}}^{2}$ as $\sqrt[3]{34^{2}}$ .

$y = \frac{\sqrt[3]{x^{2} + 102 K} \sqrt[3]{34^{2}}}{34}$

Step 3.4.8.2

Raise $34$ to the power of $2$ .

$y = \frac{\sqrt[3]{x^{2} + 102 K} \sqrt[3]{1156}}{34}$

Step 3.4.9

Simplify with factoring out.

Tap for more steps...

Step 3.4.9.1

Combine using the product rule for radicals.

$y = \frac{\sqrt[3]{(x^{2} + 102 K) \cdot 1156}}{34}$

Step 3.4.9.2

Reorder factors in $\frac{\sqrt[3]{(x^{2} + 102 K) \cdot 1156}}{34}$ .

$y = \frac{\sqrt[3]{1156 (x^{2} + 102 K)}}{34}$

Step 4

Simplify the constant of integration.

$y = \frac{\sqrt[3]{1156 (x^{2} + K)}}{34}$