Calculus Examples

$x^{3} + y^{3} = 7$

Step 1

Solve the equation as $y$ in terms of $x$ .

Tap for more steps...

Step 1.1

Subtract $x^{3}$ from both sides of the equation.

$y^{3} = 7 - x^{3}$

Step 1.2

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

$y = \sqrt[3]{7 - x^{3}}$

Step 2

Set each solution of $y$ as a function of $x$ .

$y = \sqrt[3]{7 - x^{3}} \to f (x) = \sqrt[3]{7 - x^{3}}$

Step 3

Because the $y$ variable in the equation $x^{3} + y^{3} = 7$ has a degree greater than $1$ , use implicit differentiation to solve for the derivative $\frac{d y}{d x}$ .

Tap for more steps...

Step 3.1

Differentiate both sides of the equation.

$\frac{d}{d x} (x^{3} + y^{3}) = \frac{d}{d x} (7)$

Step 3.2

Differentiate the left side of the equation.

Tap for more steps...

Step 3.2.1

Differentiate.

Tap for more steps...

Step 3.2.1.1

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

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

Step 3.2.1.2

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

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

Step 3.2.2

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

Tap for more steps...

Step 3.2.2.1

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 3.2.2.1.1

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

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

Step 3.2.2.1.2

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

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

Step 3.2.2.1.3

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

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

Step 3.2.2.2

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

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

Step 3.3

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$0$

Step 3.4

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

$3 x^{2} + 3 y^{2} y' = 0$

Step 3.5

Solve for $y'$ .

Tap for more steps...

Step 3.5.1

Subtract $3 x^{2}$ from both sides of the equation.

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

Step 3.5.2

Divide each term in $3 y^{2} y' = - 3 x^{2}$ by $3 y^{2}$ and simplify.

Tap for more steps...

Step 3.5.2.1

Divide each term in $3 y^{2} y' = - 3 x^{2}$ by $3 y^{2}$ .

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

Step 3.5.2.2

Simplify the left side.

Tap for more steps...

Step 3.5.2.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.5.2.2.1.1

Cancel the common factor.

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

Step 3.5.2.2.1.2

Rewrite the expression.

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

Step 3.5.2.2.2

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

Tap for more steps...

Step 3.5.2.2.2.1

Cancel the common factor.

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

Step 3.5.2.2.2.2

Divide $y'$ by $1$ .

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

Step 3.5.2.3

Simplify the right side.

Tap for more steps...

Step 3.5.2.3.1

Cancel the common factor of $- 3$ and $3$ .

Tap for more steps...

Step 3.5.2.3.1.1

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

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

Step 3.5.2.3.1.2

Cancel the common factors.

Tap for more steps...

Step 3.5.2.3.1.2.1

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

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

Step 3.5.2.3.1.2.2

Cancel the common factor.

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

Step 3.5.2.3.1.2.3

Rewrite the expression.

$y' = \frac{- x^{2}}{y^{2}}$

Step 3.5.2.3.2

Move the negative in front of the fraction.

$y' = - \frac{x^{2}}{y^{2}}$

Step 3.6

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

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

Step 4

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

Tap for more steps...

Step 4.1

Set the numerator equal to zero.

$x^{2} = 0$

Step 4.2

Solve the equation for $x$ .

Tap for more steps...

Step 4.2.1

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

$x = \pm \sqrt{0}$

Step 4.2.2

Simplify $\pm \sqrt{0}$ .

Tap for more steps...

Step 4.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 4.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 4.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 5

Solve the function $f (x) = \sqrt[3]{7 - x^{3}}$ at $x = 0$ .

Tap for more steps...

Step 5.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = \sqrt[3]{7 - {(0)}^{3}}$

Step 5.2

Simplify the result.

Tap for more steps...

Step 5.2.1

Raising $0$ to any positive power yields $0$ .

$f (0) = \sqrt[3]{7 - 0}$

Step 5.2.2

Multiply $- 1$ by $0$ .

$f (0) = \sqrt[3]{7 + 0}$

Step 5.2.3

Add $7$ and $0$ .

$f (0) = \sqrt[3]{7}$

Step 5.2.4

The final answer is $\sqrt[3]{7}$ .

$\sqrt[3]{7}$

Step 6

The horizontal tangent lines are $y = \sqrt[3]{7}$

$y = \sqrt[3]{7}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$y = \sqrt[3]{7}$

Decimal Form:

$y = 1.91293118 \dots$

Step 8