# Find \frac{dy}{dx} by implicit differentiation. \sqrt{3x + y} = 6 + x^2y^2 y' =

## Question:

Find {eq}\frac{dy}{dx} {/eq} by implicit differentiation.

{eq}\sqrt{3x + y} = 6 + x^2y^2 {/eq}

y' = _____

## Implicit Differentiation:

In mathematics, some equations in {eq}x {/eq} and {eq}y {/eq} do not explicitly define {eq}y {/eq} as a function {eq}x {/eq} and cannot be easily manipulated to solve for {eq}y {/eq} in terms of {eq}x {/eq}, even though such a function may exist. When this occurs, it is implied that there exists a function {eq}y = f( x) {/eq} such that the given equation is satisfied. The technique of implicit differentiation allows you to find the derivative of {eq}y {/eq} with respect to {eq}x {/eq} without having to solve the given equation for {eq}y {/eq}.

We can follow some steps for implicit differentiation:

1. Differentiate with respect to {eq}x {/eq}

2. Collect all {eq}\frac{dy}{dx} {/eq} on one side

3. Solve for {eq}\frac{dy}{dx} {/eq}.

We also use the Chain Rule in given problem:

{eq}\frac{d}{{dx}}\left[ {f\left( {g(x)} \right)} \right] = f'(g(x)).g'(x) {/eq}

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We're given {eq}\sqrt{3x + y} = 6 + x^2y^2 {/eq}

To find {eq}\frac{dy}{dx} {/eq}, we differentiate both sides of the equation with respect to... Implicit Differentiation Technique, Formula & Examples

from

Chapter 6 / Lesson 5
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Differentiation techniques include implicit differentiation. In this lesson we define the method of implicit differentiation and demonstrate this technique with numerous examples.