Calculus Examples

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

Step 1

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

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Step 1.1

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

$y = \pm \sqrt{x^{3} + 3 x^{2}}$

Step 1.2

Simplify $\pm \sqrt{x^{3} + 3 x^{2}}$ .

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Step 1.2.1

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

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Step 1.2.1.1

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

$y = \pm \sqrt{x^{2} x + 3 x^{2}}$

Step 1.2.1.2

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

$y = \pm \sqrt{x^{2} x + x^{2} \cdot 3}$

Step 1.2.1.3

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

$y = \pm \sqrt{x^{2} (x + 3)}$

Step 1.2.2

Pull terms out from under the radical.

$y = \pm x \sqrt{x + 3}$

Step 1.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.3.1

First, use the positive value of the $\pm$ to find the first solution.

$y = x \sqrt{x + 3}$

Step 1.3.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - x \sqrt{x + 3}$

Step 1.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = x \sqrt{x + 3}$

$y = - x \sqrt{x + 3}$

$y = x \sqrt{x + 3}$

$y = - x \sqrt{x + 3}$

$y = x \sqrt{x + 3}$

$y = - x \sqrt{x + 3}$

Step 2

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

$y = x \sqrt{x + 3} \to f (x) = x \sqrt{x + 3}$

$y = - x \sqrt{x + 3} \to f (x) = - x \sqrt{x + 3}$

Step 3

Because the $y$ variable in the equation $y^{2} = x^{3} + 3 x^{2}$ has a degree greater than $1$ , use implicit differentiation to solve for the derivative $\frac{d y}{d x}$ .

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Step 3.1

Differentiate both sides of the equation.

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

Step 3.2

Differentiate the left side of the equation.

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Step 3.2.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = y$ .

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Step 3.2.1.1

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

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

Step 3.2.1.2

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

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

Step 3.2.1.3

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

$2 y \frac{d}{d x} [y]$

Step 3.2.2

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

$2 y y'$

Step 3.3

Differentiate the right side of the equation.

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Step 3.3.1

Differentiate.

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Step 3.3.1.1

By the Sum Rule, the derivative of $x^{3} + 3 x^{2}$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [3 x^{2}]$ .

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

Step 3.3.1.2

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

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

Step 3.3.2

Evaluate $\frac{d}{d x} [3 x^{2}]$ .

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Step 3.3.2.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

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

Step 3.3.2.2

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

$3 x^{2} + 3 (2 x)$

Step 3.3.2.3

Multiply $2$ by $3$ .

$3 x^{2} + 6 x$

Step 3.4

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

$2 y y' = 3 x^{2} + 6 x$

Step 3.5

Divide each term in $2 y y' = 3 x^{2} + 6 x$ by $2 y$ and simplify.

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Step 3.5.1

Divide each term in $2 y y' = 3 x^{2} + 6 x$ by $2 y$ .

$\frac{2 y y'}{2 y} = \frac{3 x^{2}}{2 y} + \frac{6 x}{2 y}$

Step 3.5.2

Simplify the left side.

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Step 3.5.2.1

Cancel the common factor of $2$ .

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Step 3.5.2.1.1

Cancel the common factor.

$\frac{2 y y'}{2 y} = \frac{3 x^{2}}{2 y} + \frac{6 x}{2 y}$

Step 3.5.2.1.2

Rewrite the expression.

$\frac{y y'}{y} = \frac{3 x^{2}}{2 y} + \frac{6 x}{2 y}$

Step 3.5.2.2

Cancel the common factor of $y$ .

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Step 3.5.2.2.1

Cancel the common factor.

$\frac{y y'}{y} = \frac{3 x^{2}}{2 y} + \frac{6 x}{2 y}$

Step 3.5.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{3 x^{2}}{2 y} + \frac{6 x}{2 y}$

Step 3.5.3

Simplify the right side.

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Step 3.5.3.1

Cancel the common factor of $6$ and $2$ .

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Step 3.5.3.1.1

Factor $2$ out of $6 x$ .

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

Step 3.5.3.1.2

Cancel the common factors.

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Step 3.5.3.1.2.1

Factor $2$ out of $2 y$ .

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

Step 3.5.3.1.2.2

Cancel the common factor.

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

Step 3.5.3.1.2.3

Rewrite the expression.

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

Step 3.6

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

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

Step 4

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

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Step 4.1

Find the LCD of the terms in the equation.

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Step 4.1.1

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

$2 y, y, 1$

Step 4.1.2

Since $2 y, y, 1$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $2, 1, 1$ then find LCM for the variable part $y^{1}, y^{1}$ .

Step 4.1.3

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 4.1.4

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

$2$ is a prime number

Step 4.1.5

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

Not prime

Step 4.1.6

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

$2$

Step 4.1.7

The factor for $y^{1}$ is $y$ itself.

$y^{1} = y$

$y$ occurs $1$ time.

Step 4.1.8

The LCM of $y^{1}, y^{1}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$y$

Step 4.1.9

The LCM for $2 y, y, 1$ is the numeric part $2$ multiplied by the variable part.

$2 y$

Step 4.2

Multiply each term in $\frac{3 x^{2}}{2 y} + \frac{3 x}{y} = 0$ by $2 y$ to eliminate the fractions.

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Step 4.2.1

Multiply each term in $\frac{3 x^{2}}{2 y} + \frac{3 x}{y} = 0$ by $2 y$ .

$\frac{3 x^{2}}{2 y} (2 y) + \frac{3 x}{y} (2 y) = 0 (2 y)$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

Rewrite using the commutative property of multiplication.

$2 \frac{3 x^{2}}{2 y} y + \frac{3 x}{y} (2 y) = 0 (2 y)$

Step 4.2.2.1.2

Cancel the common factor of $2$ .

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Step 4.2.2.1.2.1

Factor $2$ out of $2 y$ .

$2 \frac{3 x^{2}}{2 (y)} y + \frac{3 x}{y} (2 y) = 0 (2 y)$

Step 4.2.2.1.2.2

Cancel the common factor.

$2 \frac{3 x^{2}}{2 y} y + \frac{3 x}{y} (2 y) = 0 (2 y)$

Step 4.2.2.1.2.3

Rewrite the expression.

$\frac{3 x^{2}}{y} y + \frac{3 x}{y} (2 y) = 0 (2 y)$

Step 4.2.2.1.3

Cancel the common factor of $y$ .

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Step 4.2.2.1.3.1

Cancel the common factor.

$\frac{3 x^{2}}{y} y + \frac{3 x}{y} (2 y) = 0 (2 y)$

Step 4.2.2.1.3.2

Rewrite the expression.

$3 x^{2} + \frac{3 x}{y} (2 y) = 0 (2 y)$

Step 4.2.2.1.4

Rewrite using the commutative property of multiplication.

$3 x^{2} + 2 \frac{3 x}{y} y = 0 (2 y)$

Step 4.2.2.1.5

Multiply $2 \frac{3 x}{y}$ .

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Step 4.2.2.1.5.1

Combine $2$ and $\frac{3 x}{y}$ .

$3 x^{2} + \frac{2 (3 x)}{y} y = 0 (2 y)$

Step 4.2.2.1.5.2

Multiply $3$ by $2$ .

$3 x^{2} + \frac{6 x}{y} y = 0 (2 y)$

Step 4.2.2.1.6

Cancel the common factor of $y$ .

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Step 4.2.2.1.6.1

Cancel the common factor.

$3 x^{2} + \frac{6 x}{y} y = 0 (2 y)$

Step 4.2.2.1.6.2

Rewrite the expression.

$3 x^{2} + 6 x = 0 (2 y)$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Multiply $0 (2 y)$ .

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Step 4.2.3.1.1

Multiply $2$ by $0$ .

$3 x^{2} + 6 x = 0 y$

Step 4.2.3.1.2

Multiply $0$ by $y$ .

$3 x^{2} + 6 x = 0$

Step 4.3

Solve the equation.

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Step 4.3.1

Factor $3 x$ out of $3 x^{2} + 6 x$ .

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Step 4.3.1.1

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

$3 x (x) + 6 x = 0$

Step 4.3.1.2

Factor $3 x$ out of $6 x$ .

$3 x (x) + 3 x (2) = 0$

Step 4.3.1.3

Factor $3 x$ out of $3 x (x) + 3 x (2)$ .

$3 x (x + 2) = 0$

Step 4.3.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x + 2 = 0$

Step 4.3.3

Set $x$ equal to $0$ .

$x = 0$

Step 4.3.4

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 4.3.4.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 4.3.4.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 4.3.5

The final solution is all the values that make $3 x (x + 2) = 0$ true.

$x = 0, - 2$

Step 5

Solve the function $f (x) = x \sqrt{x + 3}$ at $x = 0$ .

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Step 5.1

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

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

Step 5.2

Simplify the result.

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Step 5.2.1

Add $0$ and $3$ .

$f (0) = 0 \sqrt{3}$

Step 5.2.2

Multiply $0$ by $\sqrt{3}$ .

$f (0) = 0$

Step 5.2.3

The final answer is $0$ .

$0$

Step 6

Solve the function $f (x) = x \sqrt{x + 3}$ at $x = - 2$ .

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Step 6.1

Replace the variable $x$ with $- 2$ in the expression.

$f (- 2) = (- 2) \sqrt{(- 2) + 3}$

Step 6.2

Simplify the result.

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Step 6.2.1

Add $- 2$ and $3$ .

$f (- 2) = - 2 \sqrt{1}$

Step 6.2.2

Any root of $1$ is $1$ .

$f (- 2) = - 2 \cdot 1$

Step 6.2.3

Multiply $- 2$ by $1$ .

$f (- 2) = - 2$

Step 6.2.4

The final answer is $- 2$ .

$- 2$

Step 7

Solve the function $f (x) = - x \sqrt{x + 3}$ at $x = - 2$ .

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Step 7.1

Replace the variable $x$ with $- 2$ in the expression.

$f (- 2) = 2 \sqrt{(- 2) + 3}$

Step 7.2

Simplify the result.

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Step 7.2.1

Add $- 2$ and $3$ .

$f (- 2) = 2 \sqrt{1}$

Step 7.2.2

Any root of $1$ is $1$ .

$f (- 2) = 2 \cdot 1$

Step 7.2.3

Multiply $2$ by $1$ .

$f (- 2) = 2$

Step 7.2.4

The final answer is $2$ .

$2$

Step 8

The horizontal tangent lines are $y = 0, y = 0, y = - 2, y = 2$

$y = 0, y = 0, y = - 2, y = 2$

Step 9