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Define limit of a function. If $x^{2}+y^{2}+3xy=7$ , then find $(dy)/(dx)$ If $y=\cos x$ , then S.T. $y_{1}^{2}+y^{2}=1$

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Find derivatives of logarithmic functions

Full solution

Q. Define limit of a function. If

x^{2}+y^{2}+3xy=7

, then find

(dy)/(dx)

If

y=\cos x

, then S.T.

y_{1}^{2}+y^{2}=1

Identify Equation: Identify the first equation to differentiate implicitly with respect to $x$ .
Differentiate Implicitly: Differentiate both sides of $x^2 + y^2 + 3xy = 7$ with respect to $x$ . $\frac{d}{dx}(x^2) + \frac{d}{dx}(y^2) + \frac{d}{dx}(3xy) = \frac{d}{dx}(7)$ $2x + 2y\frac{dy}{dx} + 3(y + x\frac{dy}{dx}) = 0$
Solve for $\frac{dy}{dx}$ : Solve for $\frac{dy}{dx}$ . $2y\frac{dy}{dx} + 3x\frac{dy}{dx} = -2x - 3y$ $\frac{dy}{dx}(2y + 3x) = -2x - 3y$ $\frac{dy}{dx} = \frac{-2x - 3y}{2y + 3x}$
Identify Second Equation: Identify the second equation $y = \cos x$ to differentiate directly.
Differentiate Directly: Differentiate both sides of $y = \cos x$ with respect to $x$ . $\frac{dy}{dx} = -\sin x$
Evaluate at $x=1$ : Evaluate $\frac{dy}{dx}$ at $x = 1$ for $y = \cos x$ . $\frac{dy}{dx}\bigg|_{(x=1)} = -\sin(1)$
Find $y(1)$ : Use the given $y(1)^2 + y^2 = 1$ to find $y(1)$ .
$\cos(1)^2 + y^2 = 1$
$y^2 = 1 - \cos(1)^2$
$y(1) = \sqrt{1 - \cos(1)^2}$

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