Find dy/dx 4xy+1=0

$4 x y + 1 = 0$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (4 x y + 1) = \frac{d}{d x} (0)$

Step 2

Differentiate the left side of the equation.

By the Sum Rule, the derivative of $4 x y + 1$ with respect to $x$ is $\frac{d}{d x} [4 x y] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [4 x y] + \frac{d}{d x} [1]$

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

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

$4 \frac{d}{d x} [x y] + \frac{d}{d x} [1]$

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 $g (x) = y$ .

$4 (x \frac{d}{d x} [y] + y \frac{d}{d x} [x]) + \frac{d}{d x} [1]$

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

$4 (x y' + y \frac{d}{d x} [x]) + \frac{d}{d x} [1]$

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

$4 (x y' + y \cdot 1) + \frac{d}{d x} [1]$

Multiply $y$ by $1$ .

$4 (x y' + y) + \frac{d}{d x} [1]$

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

$4 (x y' + y) + 0$

Simplify.

Apply the distributive property.

$4 (x y') + 4 y + 0$

Add $4 x y' + 4 y$ and $0$ .

$4 x y' + 4 y$

Step 3

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

$0$

Step 4

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

$4 x y' + 4 y = 0$

Step 5

Solve for $y'$ .

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

$4 x y' = - 4 y$

Divide each term in $4 x y' = - 4 y$ by $4 x$ and simplify.

Divide each term in $4 x y' = - 4 y$ by $4 x$ .

$\frac{4 x y'}{4 x} = \frac{- 4 y}{4 x}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x y'}{4 x} = \frac{- 4 y}{4 x}$

Rewrite the expression.

$\frac{x y'}{x} = \frac{- 4 y}{4 x}$

Cancel the common factor of $x$ .

Cancel the common factor.

$\frac{x y'}{x} = \frac{- 4 y}{4 x}$

Divide $y'$ by $1$ .

$y' = \frac{- 4 y}{4 x}$

Simplify the right side.

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

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

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

Cancel the common factors.

Factor $4$ out of $4 x$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

Step 6

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

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

Find dy/dx 4xy+1=0

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