/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 57 Find \(\frac{d y}{d x}\), if \(2... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 57

Find \(\frac{d y}{d x}\), if \(2 x^{2}+3 x y+3 y^{2}=1\)

Short Answer

Expert verified
The derivative \( \frac{d y}{d x} = \frac{-4x - 3y}{3x + 6y} \).

Step by step solution

01

Implicit Differentiation of Each Term

The derivative of each term in the equation, using the rule of implicit differentiation, is as follows: The derivative of \(2x^2\) with respect to \(x\) is \(4x\), the derivative of \(3xy\) with respect to \(x\) is \(3y + 3x\frac{dy}{dx}\) by the product rule, the derivative of \(3y^2\) with respect to \(x\) is \(6y\frac{dy}{dx}\), as we treat \(y\) as an implicit function of \(x\), the derivative of the constant \(1\) is \(0\). As a result, we obtain the following equation: \(4x + 3y + 3x\frac{dy}{dx} + 6y\frac{dy}{dx} = 0\).
02

Isolate \(dy/dx\)

Now we can isolate \(dy/dx\) on one side of equation. We keep all terms involving \(dy/dx\) on one side and move the remaining terms to the other side. We get: \(3x\frac{dy}{dx} + 6y\frac{dy}{dx} = -4x -3y\).
03

Solving for \(dy/dx\)

To solve for \(\frac{dy}{dx}\), pull out the \(\frac{dy}{dx}\) from both terms: \(\frac{dy}{dx}(3x + 6y) = -4x -3y\). Now finally, to get \(\frac{dy}{dx}\), we divide both sides by \(3x + 6y\). This gives us: \(\frac{dy}{dx} = \frac{-4x - 3y}{3x + 6y}\).

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Most popular questions from this chapter

The length of the longest interval in which Rolle's theorem can be applied for the function \(f(x)=\left|x^{2}-a^{2}\right|\) is \((a>0)\) (a) \(2 a\) (b) \(4 a^{2}\) (c) \(a \sqrt{2}\) (d) \(a\)

If \((a+b x) e^{\frac{x}{y}}=x\), then prove that \(x^{3} \frac{d^{2} y}{d x^{2}}=\left(x \frac{d y}{d x}-y\right)^{2}\).

If \(y=c_{1} e^{x}+c_{2} e^{-x}\), then prove that \(\frac{d^{2} y}{d x^{2}}-y=0\).

\(f(x)=x^{\alpha} \sin \left(\frac{1}{x}\right), x \neq 0, f(0)=0\) satisfies conditions of Rolle's theorem on \(\left[-\frac{1}{\pi}, \frac{1}{\pi}\right]\) for \(\alpha\) equals (a) \(-1\) (b) 0 (c) \(7 / 2\) (d) \(5 / 3\)

Find the value of $$ \lim _{x \rightarrow 0}\left(\frac{1-\cos x \cos 2 x \cos 3 x}{x^{2}}\right) $$
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