Calculus Examples

$y^{3} + x y - y = 8 x^{4}$

Step 1

Set each solution of $y^{3} + x y - y$ as a function of $x$ .

$y = 8 x^{4} \to f (x) = 8 x^{4}$

Step 2

Because the $y$ variable in the equation $y^{3} + x y - y = 8 x^{4}$ has a degree greater than $1$ , use implicit differentiation to solve for the derivative $\frac{d y}{d x}$ .

Tap for more steps...

Step 2.1

Differentiate both sides of the equation.

$\frac{d}{d x} (y^{3} + x y - y) = \frac{d}{d x} (8 x^{4})$

Step 2.2

Differentiate the left side of the equation.

Tap for more steps...

Step 2.2.1

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

$\frac{d}{d x} [y^{3}] + \frac{d}{d x} [x y] + \frac{d}{d x} [- y]$

Step 2.2.2

Evaluate $\frac{d}{d x} [y^{3}]$ .

Tap for more steps...

Step 2.2.2.1

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

Tap for more steps...

Step 2.2.2.1.1

To apply the Chain Rule, set $u$ as $y$ .

$\frac{d}{d u} [u^{3}] \frac{d}{d x} [y] + \frac{d}{d x} [x y] + \frac{d}{d x} [- y]$

Step 2.2.2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 3$ .

$3 u^{2} \frac{d}{d x} [y] + \frac{d}{d x} [x y] + \frac{d}{d x} [- y]$

Step 2.2.2.1.3

Replace all occurrences of $u$ with $y$ .

$3 y^{2} \frac{d}{d x} [y] + \frac{d}{d x} [x y] + \frac{d}{d x} [- y]$

Step 2.2.2.2

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$3 y^{2} y' + \frac{d}{d x} [x y] + \frac{d}{d x} [- y]$

Step 2.2.3

Evaluate $\frac{d}{d x} [x y]$ .

Tap for more steps...

Step 2.2.3.1

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 $f (x) = x$ and $g (x) = y$ .

$3 y^{2} y' + x \frac{d}{d x} [y] + y \frac{d}{d x} [x] + \frac{d}{d x} [- y]$

Step 2.2.3.2

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$3 y^{2} y' + x y' + y \frac{d}{d x} [x] + \frac{d}{d x} [- y]$

Step 2.2.3.3

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

$3 y^{2} y' + x y' + y \cdot 1 + \frac{d}{d x} [- y]$

Step 2.2.3.4

Multiply $y$ by $1$ .

$3 y^{2} y' + x y' + y + \frac{d}{d x} [- y]$

Step 2.2.4

Evaluate $\frac{d}{d x} [- y]$ .

Tap for more steps...

Step 2.2.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- y$ with respect to $x$ is $- \frac{d}{d x} [y]$ .

$3 y^{2} y' + x y' + y - \frac{d}{d x} [y]$

Step 2.2.4.2

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$3 y^{2} y' + x y' + y - y'$

Step 2.3

Differentiate the right side of the equation.

Tap for more steps...

Step 2.3.1

Since $8$ is constant with respect to $x$ , the derivative of $8 x^{4}$ with respect to $x$ is $8 \frac{d}{d x} [x^{4}]$ .

$8 \frac{d}{d x} [x^{4}]$

Step 2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$8 (4 x^{3})$

Step 2.3.3

Multiply $4$ by $8$ .

$32 x^{3}$

Step 2.4

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

$3 y^{2} y' + x y' + y - y' = 32 x^{3}$

Step 2.5

Solve for $y'$ .

Tap for more steps...

Step 2.5.1

Subtract $y$ from both sides of the equation.

$3 y^{2} y' + x y' - y' = 32 x^{3} - y$

Step 2.5.2

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

Tap for more steps...

Step 2.5.2.1

Factor $y'$ out of $3 y^{2} y'$ .

$y' (3 y^{2}) + x (y') - y' = 32 x^{3} - y$

Step 2.5.2.2

Factor $y'$ out of $x y'$ .

$y' (3 y^{2}) + y' (x) - y' = 32 x^{3} - y$

Step 2.5.2.3

Factor $y'$ out of $- y'$ .

$y' (3 y^{2}) + y' (x) + y' \cdot - 1 = 32 x^{3} - y$

Step 2.5.2.4

Factor $y'$ out of $y' (3 y^{2}) + y' (x)$ .

$y' (3 y^{2} + x) + y' \cdot - 1 = 32 x^{3} - y$

Step 2.5.2.5

Factor $y'$ out of $y' (3 y^{2} + x) + y' \cdot - 1$ .

$y' (3 y^{2} + x - 1) = 32 x^{3} - y$

Step 2.5.3

Divide each term in $y' (3 y^{2} + x - 1) = 32 x^{3} - y$ by $3 y^{2} + x - 1$ and simplify.

Tap for more steps...

Step 2.5.3.1

Divide each term in $y' (3 y^{2} + x - 1) = 32 x^{3} - y$ by $3 y^{2} + x - 1$ .

$\frac{y' (3 y^{2} + x - 1)}{3 y^{2} + x - 1} = \frac{32 x^{3}}{3 y^{2} + x - 1} + \frac{- y}{3 y^{2} + x - 1}$

Step 2.5.3.2

Simplify the left side.

Tap for more steps...

Step 2.5.3.2.1

Cancel the common factor of $3 y^{2} + x - 1$ .

Tap for more steps...

Step 2.5.3.2.1.1

Cancel the common factor.

$\frac{y' (3 y^{2} + x - 1)}{3 y^{2} + x - 1} = \frac{32 x^{3}}{3 y^{2} + x - 1} + \frac{- y}{3 y^{2} + x - 1}$

Step 2.5.3.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{32 x^{3}}{3 y^{2} + x - 1} + \frac{- y}{3 y^{2} + x - 1}$

Step 2.5.3.3

Simplify the right side.

Tap for more steps...

Step 2.5.3.3.1

Combine the numerators over the common denominator.

$y' = \frac{32 x^{3} - y}{3 y^{2} + x - 1}$

Step 2.6

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{32 x^{3} - y}{3 y^{2} + x - 1}$

Step 3

Set the derivative equal to $0$ then solve the equation $\frac{32 x^{3} - y}{3 y^{2} + x - 1} = 0$ .

Tap for more steps...

Step 3.1

Set the numerator equal to zero.

$32 x^{3} - y = 0$

Step 3.2

Solve the equation for $x$ .

Tap for more steps...

Step 3.2.1

Add $y$ to both sides of the equation.

$32 x^{3} = y$

Step 3.2.2

Divide each term in $32 x^{3} = y$ by $32$ and simplify.

Tap for more steps...

Step 3.2.2.1

Divide each term in $32 x^{3} = y$ by $32$ .

$\frac{32 x^{3}}{32} = \frac{y}{32}$

Step 3.2.2.2

Simplify the left side.

Tap for more steps...

Step 3.2.2.2.1

Cancel the common factor of $32$ .

Tap for more steps...

Step 3.2.2.2.1.1

Cancel the common factor.

$\frac{32 x^{3}}{32} = \frac{y}{32}$

Step 3.2.2.2.1.2

Divide $x^{3}$ by $1$ .

$x^{3} = \frac{y}{32}$

Step 3.2.3

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

$x = \sqrt[3]{\frac{y}{32}}$

Step 3.2.4

Simplify $\sqrt[3]{\frac{y}{32}}$ .

Tap for more steps...

Step 3.2.4.1

Rewrite $\frac{y}{32}$ as ${(\frac{1}{2})}^{3} \frac{y}{4}$ .

Tap for more steps...

Step 3.2.4.1.1

Factor the perfect power $1^{3}$ out of $y$ .

$x = \sqrt[3]{\frac{1^{3} y}{32}}$

Step 3.2.4.1.2

Factor the perfect power $2^{3}$ out of $32$ .

$x = \sqrt[3]{\frac{1^{3} y}{2^{3} \cdot 4}}$

Step 3.2.4.1.3

Rearrange the fraction $\frac{1^{3} y}{2^{3} \cdot 4}$ .

$x = \sqrt[3]{{(\frac{1}{2})}^{3} \frac{y}{4}}$

Step 3.2.4.2

Pull terms out from under the radical.

$x = \frac{1}{2} \sqrt[3]{\frac{y}{4}}$

Step 3.2.4.3

Rewrite $\sqrt[3]{\frac{y}{4}}$ as $\frac{\sqrt[3]{y}}{\sqrt[3]{4}}$ .

$x = \frac{1}{2} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{4}}$

Step 3.2.4.4

Multiply $\frac{\sqrt[3]{y}}{\sqrt[3]{4}}$ by $\frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$ .

$x = \frac{1}{2} (\frac{\sqrt[3]{y}}{\sqrt[3]{4}} \cdot \frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}})$

Step 3.2.4.5

Combine fractions.

Tap for more steps...

Step 3.2.4.5.1

Combine and simplify the denominator.

Tap for more steps...

Step 3.2.4.5.1.1

Multiply $\frac{\sqrt[3]{y}}{\sqrt[3]{4}}$ by $\frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$ .

$x = \frac{1}{2} \cdot \frac{\sqrt[3]{y} {\sqrt[3]{4}}^{2}}{\sqrt[3]{4} {\sqrt[3]{4}}^{2}}$

Step 3.2.4.5.1.2

Raise $\sqrt[3]{4}$ to the power of $1$ .

$x = \frac{1}{2} \cdot \frac{\sqrt[3]{y} {\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{1} {\sqrt[3]{4}}^{2}}$

Step 3.2.4.5.1.3

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

$x = \frac{1}{2} \cdot \frac{\sqrt[3]{y} {\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{1 + 2}}$

Step 3.2.4.5.1.4

Add $1$ and $2$ .

$x = \frac{1}{2} \cdot \frac{\sqrt[3]{y} {\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{3}}$

Step 3.2.4.5.1.5

Rewrite ${\sqrt[3]{4}}^{3}$ as $4$ .

Tap for more steps...

Step 3.2.4.5.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4}$ as $4^{\frac{1}{3}}$ .

$x = \frac{1}{2} \cdot \frac{\sqrt[3]{y} {\sqrt[3]{4}}^{2}}{{(4^{\frac{1}{3}})}^{3}}$

Step 3.2.4.5.1.5.2

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

$x = \frac{1}{2} \cdot \frac{\sqrt[3]{y} {\sqrt[3]{4}}^{2}}{4^{\frac{1}{3} \cdot 3}}$

Step 3.2.4.5.1.5.3

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

$x = \frac{1}{2} \cdot \frac{\sqrt[3]{y} {\sqrt[3]{4}}^{2}}{4^{\frac{3}{3}}}$

Step 3.2.4.5.1.5.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.2.4.5.1.5.4.1

Cancel the common factor.

$x = \frac{1}{2} \cdot \frac{\sqrt[3]{y} {\sqrt[3]{4}}^{2}}{4^{\frac{3}{3}}}$

Step 3.2.4.5.1.5.4.2

Rewrite the expression.

$x = \frac{1}{2} \cdot \frac{\sqrt[3]{y} {\sqrt[3]{4}}^{2}}{4^{1}}$

Step 3.2.4.5.1.5.5

Evaluate the exponent.

$x = \frac{1}{2} \cdot \frac{\sqrt[3]{y} {\sqrt[3]{4}}^{2}}{4}$

Step 3.2.4.5.2

Combine.

$x = \frac{1 (\sqrt[3]{y} {\sqrt[3]{4}}^{2})}{2 \cdot 4}$

Step 3.2.4.5.3

Multiply.

Tap for more steps...

Step 3.2.4.5.3.1

Multiply $\sqrt[3]{y}$ by $1$ .

$x = \frac{\sqrt[3]{y} {\sqrt[3]{4}}^{2}}{2 \cdot 4}$

Step 3.2.4.5.3.2

Multiply $2$ by $4$ .

$x = \frac{\sqrt[3]{y} {\sqrt[3]{4}}^{2}}{8}$

Step 3.2.4.6

Simplify the numerator.

Tap for more steps...

Step 3.2.4.6.1

Rewrite ${\sqrt[3]{4}}^{2}$ as $\sqrt[3]{4^{2}}$ .

$x = \frac{\sqrt[3]{y} \sqrt[3]{4^{2}}}{8}$

Step 3.2.4.6.2

Raise $4$ to the power of $2$ .

$x = \frac{\sqrt[3]{y} \sqrt[3]{16}}{8}$

Step 3.2.4.6.3

Rewrite $16$ as $2^{3} \cdot 2$ .

Tap for more steps...

Step 3.2.4.6.3.1

Factor $8$ out of $16$ .

$x = \frac{\sqrt[3]{y} \sqrt[3]{8 (2)}}{8}$

Step 3.2.4.6.3.2

Rewrite $8$ as $2^{3}$ .

$x = \frac{\sqrt[3]{y} \sqrt[3]{2^{3} \cdot 2}}{8}$

Step 3.2.4.6.4

Pull terms out from under the radical.

$x = \frac{\sqrt[3]{y} \cdot 2 \sqrt[3]{2}}{8}$

Step 3.2.4.6.5

Combine using the product rule for radicals.

$x = \frac{2 \sqrt[3]{y \cdot 2}}{8}$

Step 3.2.4.7

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 3.2.4.7.1

Cancel the common factor of $2$ and $8$ .

Tap for more steps...

Step 3.2.4.7.1.1

Factor $2$ out of $2 \sqrt[3]{y \cdot 2}$ .

$x = \frac{2 (\sqrt[3]{y \cdot 2})}{8}$

Step 3.2.4.7.1.2

Cancel the common factors.

Tap for more steps...

Step 3.2.4.7.1.2.1

Factor $2$ out of $8$ .

$x = \frac{2 \sqrt[3]{y \cdot 2}}{2 \cdot 4}$

Step 3.2.4.7.1.2.2

Cancel the common factor.

$x = \frac{2 \sqrt[3]{y \cdot 2}}{2 \cdot 4}$

Step 3.2.4.7.1.2.3

Rewrite the expression.

$x = \frac{\sqrt[3]{y \cdot 2}}{4}$

Step 3.2.4.7.2

Reorder factors in $\frac{\sqrt[3]{y \cdot 2}}{4}$ .

$x = \frac{\sqrt[3]{2 y}}{4}$

Step 4

Solve the function $f (x) = 8 x^{4}$ at $x = \frac{\sqrt[3]{2 y}}{4}$ .

Tap for more steps...

Step 4.1

Replace the variable $x$ with $\frac{\sqrt[3]{2 y}}{4}$ in the expression.

$f (\frac{\sqrt[3]{2 y}}{4}) = 8 {(\frac{\sqrt[3]{2 y}}{4})}^{4}$

Step 4.2

Simplify the result.

Tap for more steps...

Step 4.2.1

Apply the product rule to $\frac{\sqrt[3]{2 y}}{4}$ .

$f (\frac{\sqrt[3]{2 y}}{4}) = 8 (\frac{{\sqrt[3]{2 y}}^{4}}{4^{4}})$

Step 4.2.2

Simplify the numerator.

Tap for more steps...

Step 4.2.2.1

Rewrite ${\sqrt[3]{2 y}}^{4}$ as $\sqrt[3]{{(2 y)}^{4}}$ .

$f (\frac{\sqrt[3]{2 y}}{4}) = 8 (\frac{\sqrt[3]{{(2 y)}^{4}}}{4^{4}})$

Step 4.2.2.2

Apply the product rule to $2 y$ .

$f (\frac{\sqrt[3]{2 y}}{4}) = 8 (\frac{\sqrt[3]{2^{4} y^{4}}}{4^{4}})$

Step 4.2.2.3

Raise $2$ to the power of $4$ .

$f (\frac{\sqrt[3]{2 y}}{4}) = 8 (\frac{\sqrt[3]{16 y^{4}}}{4^{4}})$

Step 4.2.2.4

Rewrite $16 y^{4}$ as ${(2 y)}^{3} \cdot (2 y)$ .

Tap for more steps...

Step 4.2.2.4.1

Factor $8$ out of $16$ .

$f (\frac{\sqrt[3]{2 y}}{4}) = 8 (\frac{\sqrt[3]{8 (2) y^{4}}}{4^{4}})$

Step 4.2.2.4.2

Rewrite $8$ as $2^{3}$ .

$f (\frac{\sqrt[3]{2 y}}{4}) = 8 (\frac{\sqrt[3]{2^{3} \cdot (2 y^{4})}}{4^{4}})$

Step 4.2.2.4.3

Factor out $y^{3}$ .

$f (\frac{\sqrt[3]{2 y}}{4}) = 8 (\frac{\sqrt[3]{2^{3} \cdot (2 (y^{3} y))}}{4^{4}})$

Step 4.2.2.4.4

Move $2$ .

$f (\frac{\sqrt[3]{2 y}}{4}) = 8 (\frac{\sqrt[3]{2^{3} y^{3} \cdot (2 y)}}{4^{4}})$

Step 4.2.2.4.5

Rewrite $2^{3} y^{3}$ as ${(2 y)}^{3}$ .

$f (\frac{\sqrt[3]{2 y}}{4}) = 8 (\frac{\sqrt[3]{{(2 y)}^{3} \cdot (2 y)}}{4^{4}})$

Step 4.2.2.4.6

Add parentheses.

$f (\frac{\sqrt[3]{2 y}}{4}) = 8 (\frac{\sqrt[3]{{(2 y)}^{3} \cdot (2 y)}}{4^{4}})$

Step 4.2.2.5

Pull terms out from under the radical.

$f (\frac{\sqrt[3]{2 y}}{4}) = 8 (\frac{2 y \sqrt[3]{2 y}}{4^{4}})$

Step 4.2.3

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 4.2.3.1

Raise $4$ to the power of $4$ .

$f (\frac{\sqrt[3]{2 y}}{4}) = 8 (\frac{2 y \sqrt[3]{2 y}}{256})$

Step 4.2.3.2

Cancel the common factor of $8$ .

Tap for more steps...

Step 4.2.3.2.1

Factor $8$ out of $256$ .

$f (\frac{\sqrt[3]{2 y}}{4}) = 8 (\frac{2 y \sqrt[3]{2 y}}{8 (32)})$

Step 4.2.3.2.2

Cancel the common factor.

$f (\frac{\sqrt[3]{2 y}}{4}) = 8 (\frac{2 y \sqrt[3]{2 y}}{8 \cdot 32})$

Step 4.2.3.2.3

Rewrite the expression.

$f (\frac{\sqrt[3]{2 y}}{4}) = \frac{2 y \sqrt[3]{2 y}}{32}$

Step 4.2.3.3

Cancel the common factor of $2$ and $32$ .

Tap for more steps...

Step 4.2.3.3.1

Factor $2$ out of $2 y \sqrt[3]{2 y}$ .

$f (\frac{\sqrt[3]{2 y}}{4}) = \frac{2 (y \sqrt[3]{2 y})}{32}$

Step 4.2.3.3.2

Cancel the common factors.

Tap for more steps...

Step 4.2.3.3.2.1

Factor $2$ out of $32$ .

$f (\frac{\sqrt[3]{2 y}}{4}) = \frac{2 (y \sqrt[3]{2 y})}{2 \cdot 16}$

Step 4.2.3.3.2.2

Cancel the common factor.

$f (\frac{\sqrt[3]{2 y}}{4}) = \frac{2 (y \sqrt[3]{2 y})}{2 \cdot 16}$

Step 4.2.3.3.2.3

Rewrite the expression.

$f (\frac{\sqrt[3]{2 y}}{4}) = \frac{y \sqrt[3]{2 y}}{16}$

Step 4.2.4

The final answer is $\frac{y \sqrt[3]{2 y}}{16}$ .

$\frac{y \sqrt[3]{2 y}}{16}$

Step 5

The horizontal tangent lines are $y = \frac{y \sqrt[3]{2 y}}{16}$

$y = \frac{y \sqrt[3]{2 y}}{16}$

Step 6