Answered: How is the Chain Rule applied when…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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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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