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Chapter 2: Q. 69 (page 211)

Each of the equations in Exercises 69鈥�80 defines y as an implicit function of x. Use implicit differentiation (without solving for y first) to find $\frac{d y}{d x}$
$4 x^{2} - y^{2} = 9$

Short Answer

Expert verified

$\frac{d y}{d x} = \frac{4 x}{y}$

Step by step solution

01

Step 1. Given Information:

Given equation: $4 x^{2} - y^{2} = 9$

We want to find $\frac{d y}{d x}$ defines y as an implicit function of x by use implicit differentiation.

02

Step 2. Solution:

Differentiate both sides w.r.t. x

$\frac{d}{d x} (4 x^{2} - y^{2}) = \frac{d}{d x} (9) \frac{d}{d x} 4 x^{2} - \frac{d}{d x} y^{2} = \frac{d}{d x} 9 8 x - 2 y 路 \frac{d y}{d x} = 0 2 y 路 \frac{d y}{d x} = 8 x \frac{d y}{d x} = \frac{8 x}{2 y} \frac{d y}{d x} = \frac{4 x}{y}$

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Most popular questions from this chapter

State the chain rule for differentiating a composition $g (h (x))$ of two functions expressed

(a) in 鈥減rime鈥� notation and

(b) in Leibniz notation.

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Find the derivatives of the functions in Exercises 21鈥�46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating.

$f (x) = {(x^{\frac{1}{3}} - 2 x)}^{- 1}$

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$f (x) = \frac{2}{1 - x}$

Find the derivatives of the functions in Exercises 21鈥�46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating.

$f (x) = \sqrt{2 - \sqrt{3 x + 1}}$

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