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Use implicit differentiation to find dz/dx and dz/dy. x^2 + 4y^2 + 7z^2 = 4.

Question:

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

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

Implicit Differentiation:

You have to find the implicit differentiation for the the given function. First you have to find implicit differentiation with respect to x , in that case you will take other constant y as constant and do normal differentiate. You have to find implicit differentiation with respect to y , in that case you will take other constant x as constant and do normal differentiate.

Answer and Explanation:

You have the given function $$\begin{align*} {x^2} + 4{y^2} + 7{z^2} &= 4\\ 7{z^2} &= 4 - {x^2} - 4{y^2}\\ {z^2} &= \frac{4}{7} - \frac{{{x^2}}}{7} - \frac{{4{y^2}}}{7}. \end{align*} $$

Find implicit differentiation with respect to x and you have $$\begin{align*} {z^2} &= \frac{4}{7} - \frac{{{x^2}}}{7} - \frac{{4{y^2}}}{7}\\ \frac{d}{{dx}}\left[ {{z^2}} \right] &= \frac{d}{{dx}}\left[ {\frac{4}{7} - \frac{{{x^2}}}{7} - \frac{{4{y^2}}}{7}} \right]\\ 2z \cdot \frac{{dz}}{{dx}} &= - \frac{{2x}}{7}\\ \frac{{dz}}{{dx}} &= - \frac{x}{{7z}}\\ \frac{{dz}}{{dx}} &= - \frac{x}{{7\sqrt {\frac{4}{7} - \frac{{{x^2}}}{7} - \frac{{4{y^2}}}{7}} }}. \end{align*} $$

Again, find implicit differentiation with respect to x and you have $$\begin{align*} {z^2} &= \frac{4}{7} - \frac{{{x^2}}}{7} - \frac{{4{y^2}}}{7}\\ \frac{d}{{dy}}\left[ {{z^2}} \right] &= \frac{d}{{dy}}\left[ {\frac{4}{7} - \frac{{{x^2}}}{7} - \frac{{4{y^2}}}{7}} \right]\\ 2z \cdot \frac{{dz}}{{dy}} &= - \frac{{8y}}{7}\\ \frac{{dz}}{{dx}} &= - \frac{{4y}}{{7z}}\\ \frac{{dz}}{{dx}} &= - \frac{{4y}}{{7\sqrt {\frac{4}{7} - \frac{{{x^2}}}{7} - \frac{{4{y^2}}}{7}} }}. \end{align*} $$


Learn more about this topic:

How to Find Derivatives of Implicit Functions

from Math 104: Calculus

Chapter 9 / Lesson 11
9.2K

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