Calculus Examples

$- 1 = 5 x^{2} + x y$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (- 1) = \frac{d}{d x} (5 x^{2} + x y)$

Step 2

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$0$

Step 3

Differentiate the right side of the equation.

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Step 3.1

By the Sum Rule, the derivative of $5 x^{2} + x y$ with respect to $x$ is $\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [x y]$ .

$\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [x y]$

Step 3.2

Evaluate $\frac{d}{d x} [5 x^{2}]$ .

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Step 3.2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{2}$ with respect to $x$ is $5 \frac{d}{d x} [x^{2}]$ .

$5 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [x y]$

Step 3.2.2

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

$5 (2 x) + \frac{d}{d x} [x y]$

Step 3.2.3

Multiply $2$ by $5$ .

$10 x + \frac{d}{d x} [x y]$

Step 3.3

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

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Step 3.3.1

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 $g (x) = y$ .

$10 x + x \frac{d}{d x} [y] + y \frac{d}{d x} [x]$

Step 3.3.2

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

$10 x + x y' + y \frac{d}{d x} [x]$

Step 3.3.3

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

$10 x + x y' + y \cdot 1$

Step 3.3.4

Multiply $y$ by $1$ .

$10 x + x y' + y$

Step 3.4

Reorder terms.

$x y' + 10 x + y$

Step 4

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

$0 = x y' + 10 x + y$

Step 5

Solve for $y'$ .

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Step 5.1

Rewrite the equation as $x y' + 10 x + y = 0$ .

$x y' + 10 x + y = 0$

Step 5.2

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

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Step 5.2.1

Subtract $10 x$ from both sides of the equation.

$x y' + y = - 10 x$

Step 5.2.2

Subtract $y$ from both sides of the equation.

$x y' = - 10 x - y$

Step 5.3

Divide each term in $x y' = - 10 x - y$ by $x$ and simplify.

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Step 5.3.1

Divide each term in $x y' = - 10 x - y$ by $x$ .

$\frac{x y'}{x} = \frac{- 10 x}{x} + \frac{- y}{x}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $x$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{x y'}{x} = \frac{- 10 x}{x} + \frac{- y}{x}$

Step 5.3.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{- 10 x}{x} + \frac{- y}{x}$

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

Cancel the common factor of $x$ .

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Step 5.3.3.1.1.1

Cancel the common factor.

$y' = \frac{- 10 x}{x} + \frac{- y}{x}$

Step 5.3.3.1.1.2

Divide $- 10$ by $1$ .

$y' = - 10 + \frac{- y}{x}$

Step 5.3.3.1.2

Move the negative in front of the fraction.

$y' = - 10 - \frac{y}{x}$

Step 6

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

$\frac{d y}{d x} = - 10 - \frac{y}{x}$