Use implicit differentiation to find partial z/partial x and partial z/partial y. x^2 + 4y^2 +...


Use implicit differentiation to find {eq}\displaystyle \frac{\partial z}{\partial x} \ \ and \ \ \frac{\partial z}{\partial y}\ {/eq}.

{eq}x^2 + 4y^2 + 5z^2 = 7 {/eq}

Implicit Partial Differentiation:

Most of the problems we do in differentiation are explicit differentiation. Implicit differentiation, on the other hand, allows us to find {eq}\displaystyle \frac{dy}{dx} {/eq} without solving the equation for y.

In the following problem, we use formulae like:

  • {eq}\displaystyle \frac{\partial(x^a)}{\partial x} = ax^{a-1} {/eq}

Answer and Explanation: 1

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{eq}\displaystyle x^2 + 4y^2 + 5z^2 = 7 {/eq}

On partial differentiation with respect to x, we get

{eq}\displaystyle \begin{align} \frac...

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Implicit Differentiation Technique, Formula & Examples


Chapter 6 / Lesson 5

Explicit differentiation is used when 'y' is isolated, whereas implicit differentiation can be used similar to the chain rule when 'y' is not isolated. Learn more about implicit differentiation through examples of formulas and graphs.

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