Calculus Examples

$x^{3} + 2 x^{2} y - 4 y = 7$

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

Differentiate.

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Step 2.1.1

By the Sum Rule, the derivative of $x^{3} + 2 x^{2} y - 4 y$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [2 x^{2} y] + \frac{d}{d x} [- 4 y]$ .

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [2 x^{2} y] + \frac{d}{d x} [- 4 y]$

Step 2.1.2

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

$3 x^{2} + \frac{d}{d x} [2 x^{2} y] + \frac{d}{d x} [- 4 y]$

Step 2.2

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

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Step 2.2.1

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

$3 x^{2} + 2 \frac{d}{d x} [x^{2} y] + \frac{d}{d x} [- 4 y]$

Step 2.2.2

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 $f (x) = x^{2}$ and $g (x) = y$ .

$3 x^{2} + 2 (x^{2} \frac{d}{d x} [y] + y \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [- 4 y]$

Step 2.2.3

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

$3 x^{2} + 2 (x^{2} y' + y \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [- 4 y]$

Step 2.2.4

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

$3 x^{2} + 2 (x^{2} y' + y (2 x)) + \frac{d}{d x} [- 4 y]$

Step 2.2.5

Move $2$ to the left of $y$ .

$3 x^{2} + 2 (x^{2} y' + 2 y x) + \frac{d}{d x} [- 4 y]$

Step 2.3

Evaluate $\frac{d}{d x} [- 4 y]$ .

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Step 2.3.1

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

$3 x^{2} + 2 (x^{2} y' + 2 y x) - 4 \frac{d}{d x} [y]$

Step 2.3.2

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

$3 x^{2} + 2 (x^{2} y' + 2 y x) - 4 y'$

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

$3 x^{2} + 2 (x^{2} y') + 2 (2 y x) - 4 y'$

Step 2.4.2

Multiply $2$ by $2$ .

$3 x^{2} + 2 x^{2} y' + 4 y x - 4 y'$

Step 2.4.3

Reorder terms.

$3 x^{2} + 2 x^{2} y' + 4 x y - 4 y'$

Step 3

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

$0$

Step 4

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

$3 x^{2} + 2 x^{2} y' + 4 x y - 4 y' = 0$

Step 5

Solve for $y'$ .

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Step 5.1

Move all terms not containing $y'$ to the right side of the equation.

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Step 5.1.1

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

$2 x^{2} y' + 4 x y - 4 y' = - 3 x^{2}$

Step 5.1.2

Subtract $4 x y$ from both sides of the equation.

$2 x^{2} y' - 4 y' = - 3 x^{2} - 4 x y$

Step 5.2

Factor $2 y'$ out of $2 x^{2} y' - 4 y'$ .

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Step 5.2.1

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

$2 y' x^{2} - 4 y' = - 3 x^{2} - 4 x y$

Step 5.2.2

Factor $2 y'$ out of $- 4 y'$ .

$2 y' x^{2} + 2 y' \cdot - 2 = - 3 x^{2} - 4 x y$

Step 5.2.3

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

$2 y' (x^{2} - 2) = - 3 x^{2} - 4 x y$

Step 5.3

Divide each term in $2 y' (x^{2} - 2) = - 3 x^{2} - 4 x y$ by $2 (x^{2} - 2)$ and simplify.

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Step 5.3.1

Divide each term in $2 y' (x^{2} - 2) = - 3 x^{2} - 4 x y$ by $2 (x^{2} - 2)$ .

$\frac{2 y' (x^{2} - 2)}{2 (x^{2} - 2)} = \frac{- 3 x^{2}}{2 (x^{2} - 2)} + \frac{- 4 x y}{2 (x^{2} - 2)}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $2$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{2 y' (x^{2} - 2)}{2 (x^{2} - 2)} = \frac{- 3 x^{2}}{2 (x^{2} - 2)} + \frac{- 4 x y}{2 (x^{2} - 2)}$

Step 5.3.2.1.2

Rewrite the expression.

$\frac{y' (x^{2} - 2)}{x^{2} - 2} = \frac{- 3 x^{2}}{2 (x^{2} - 2)} + \frac{- 4 x y}{2 (x^{2} - 2)}$

Step 5.3.2.2

Cancel the common factor of $x^{2} - 2$ .

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Step 5.3.2.2.1

Cancel the common factor.

$\frac{y' (x^{2} - 2)}{x^{2} - 2} = \frac{- 3 x^{2}}{2 (x^{2} - 2)} + \frac{- 4 x y}{2 (x^{2} - 2)}$

Step 5.3.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{- 3 x^{2}}{2 (x^{2} - 2)} + \frac{- 4 x y}{2 (x^{2} - 2)}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Simplify each term.

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Step 5.3.3.1.1

Move the negative in front of the fraction.

$y' = - \frac{3 x^{2}}{2 (x^{2} - 2)} + \frac{- 4 x y}{2 (x^{2} - 2)}$

Step 5.3.3.1.2

Cancel the common factor of $- 4$ and $2$ .

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Step 5.3.3.1.2.1

Factor $2$ out of $- 4 x y$ .

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

Step 5.3.3.1.2.2

Cancel the common factors.

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Step 5.3.3.1.2.2.1

Cancel the common factor.

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

Step 5.3.3.1.2.2.2

Rewrite the expression.

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

Step 5.3.3.1.3

Move the negative in front of the fraction.

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

Step 5.3.3.2

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

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

Step 5.3.3.3

Write each expression with a common denominator of $2 (x^{2} - 2)$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.3.3.3.1

Multiply $\frac{2 x y}{x^{2} - 2}$ by $\frac{2}{2}$ .

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

Step 5.3.3.3.2

Reorder the factors of $(x^{2} - 2) \cdot 2$ .

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

Step 5.3.3.4

Combine the numerators over the common denominator.

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

Step 5.3.3.5

Simplify the numerator.

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Step 5.3.3.5.1

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

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Step 5.3.3.5.1.1

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

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

Step 5.3.3.5.1.2

Factor $x$ out of $- 2 x y \cdot 2$ .

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

Step 5.3.3.5.1.3

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

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

Step 5.3.3.5.2

Multiply $2$ by $- 2$ .

$y' = \frac{x (- 3 x - 4 y)}{2 (x^{2} - 2)}$

Step 5.3.3.6

Simplify with factoring out.

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Step 5.3.3.6.1

Factor $- 1$ out of $- 3 x$ .

$y' = \frac{x (- (3 x) - 4 y)}{2 (x^{2} - 2)}$

Step 5.3.3.6.2

Factor $- 1$ out of $- 4 y$ .

$y' = \frac{x (- (3 x) - (4 y))}{2 (x^{2} - 2)}$

Step 5.3.3.6.3

Factor $- 1$ out of $- (3 x) - (4 y)$ .

$y' = \frac{x (- (3 x + 4 y))}{2 (x^{2} - 2)}$

Step 5.3.3.6.4

Simplify the expression.

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Step 5.3.3.6.4.1

Rewrite $- (3 x + 4 y)$ as $- 1 (3 x + 4 y)$ .

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

Step 5.3.3.6.4.2

Move the negative in front of the fraction.

$y' = - \frac{x (3 x + 4 y)}{2 (x^{2} - 2)}$

Step 6

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

$\frac{d y}{d x} = - \frac{x (3 x + 4 y)}{2 (x^{2} - 2)}$