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. d/dx(x^2) + d/dx(y^2) + d/dx(3xy) = d/dx(7) 2x + 2y(dy/dx) + 3(y + x(dy/dx)) = 0Solve for dy/dx: Solve for dy/dx. 2y(dy/dx) + 3x(dy/dx) = -2x - 3y (dy/dx)(2y + 3x) = -2x - 3y dy/dx = (-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. dy/dx = -sin xEvaluate at x=1: Evaluate dy/dx at x = 1 for y = cos x. dy/dx|_(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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Find derivatives of logarithmic functions

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Q. Define limit of a function. If

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

, then find

(dy)/(dx)

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 $y(1)$ .
$\cos(1)^2 + y^2 = 1$
$y^2 = 1 - \cos(1)^2$
$y(1) = \sqrt{1 - \cos(1)^2}$