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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)^(2)=1

$37$ . Define limit of a function. $\newline$ $1$ . If $x^{2}+y^{2}+3 x y=7$ , then find $\frac{d y}{d x}$ $\newline$ $2$ . If $y=\cos x$ , then S.T. $y_{1}^{2}+y_{2}^{2}=1$

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Find derivatives of logarithmic functions

Full solution

Q.

37

. Define limit of a function.

\newline

1

. If

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

, then find

\frac{d y}{d x}

\newline

2

. If

y=\cos x

, then S.T.

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

Identify Equation: Identify the first equation to differentiate. $\newline$ $x^2 + y^2 + 3xy = 7$ $\newline$ Differentiate both sides with respect to $x$ . $\newline$ $\frac{d}{dx}(x^2) + \frac{d}{dx}(y^2) + \frac{d}{dx}(3xy) = \frac{d}{dx}(7)$
Differentiate with Respect: Apply the derivative rules. $\newline$ $2x + 2y\frac{dy}{dx} + 3(x\frac{dy}{dx} + y) = 0$
Apply Derivative Rules: Solve for $\frac{dy}{dx}$ . $2y\frac{dy}{dx} + 3x\frac{dy}{dx} = -2x - 3y$ $\frac{dy}{dx}(2y + 3x) = -2x - 3y$ $\frac{dy}{dx} = \frac{-2x - 3y}{2y + 3x}$
Solve for $\frac{dy}{dx}$ : Substitute $y = \cos(x)$ into the second equation. $\newline$ $y_1^2 + y_2^2 = \cos(x)^2 + \left(\frac{dy}{dx}\right)^2$
Substitute $y = \cos(x)$ : Differentiate $y = \cos(x)$ to find $y_2$ . $\newline$ $y_2 = \frac{dy}{dx} = -\sin(x)$
Find $y_2$ : Substitute $y_2$ into the equation. $\newline$ $y_1^2 + (-\sin(x))^2 = \cos(x)^2 + \sin(x)^2$
Substitute $y_2$ : Simplify the equation. $\cos(x)^2 + \sin(x)^2 = 1$
Simplify Equation: Check if the identity is true. $\newline$ Since $\cos(x)^2 + \sin(x)^2$ is always equal to $1$ , the identity holds.

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