Answered: How is the Chain Rule applied when…

College Algebra (MindTap Course List)

12th Edition

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:R. David Gustafson, Jeff Hughes

Chapter6: Linear Systems

Section6.2: Guassian Elimination And Matrix Methods

Problem 84E: Explain the differences between Gaussian elimination and Gauss-Jordan elimination.

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Question

How is the Chain Rule applied when finding dydx implicitly?

Expert Solution

Step 1

According to the question, we have to explain the chain rule when used in the implict function.

Implict function can be defined as the function in which the dependent variable can be written explicitly in terms of the independent variable.

For example, $x^{2} + y^{2} = 1$

The chain rule for differentiation is given by,

$\frac{d}{d x} (f (x) \cdot g (x)) = f (x) \cdot g' (x) + g (x) \cdot f' (x)$ .

Step 2

Let a conic equation $x^{2} + x y + y^{2} = 0$ to be differentiated with respect to x that is $\frac{d y}{d x}$ .

So, solving the above equation to find the value of $\frac{d y}{d x}$ .

The equation is $x^{2} + x y + y^{2} = 0$

Now, differentiating the above equation with respect to x, we get,

$2 x + x \cdot \frac{d y}{d x} + y \cdot 1 + 2 y \cdot \frac{d y}{d x} = 0 2 x + x \frac{d y}{d x} + y + 2 y \frac{d y}{d x} = 0 \frac{d y}{d x} (x + 2 y) = - 2 x - y \frac{d y}{d x} = \frac{- (2 x + y)}{(x + 2 y)}$

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