Question

question 20
find \\( \frac{dy}{dx} \\) when \\( y^2 - 5x = 11 \\).
\\( \bigcirc \\) \\( \frac{dy}{dx} = 2y - 5 \\)
\\( \bigcirc \\) no correct answer choice is given.
\\( \bigcirc \\) \\( \frac{dy}{dx} = 11 \\)
\\( \bigcirc \\) \\( \frac{dy}{dx} = \frac{5x}{2y} \\)
\\( \bigcirc \\) \\( \frac{dy}{dx} = \frac{5}{2y} \\)

Explanation:

Step1: Differentiate both sides w.r. to x

Differentiate \( y^2 - 5x = 11 \) with respect to \( x \). Using chain rule for \( y^2 \) (derivative is \( 2y\frac{dy}{dx} \)) and power rule for \( -5x \) (derivative is \( -5 \)), and derivative of constant 11 is 0. So we get: \( 2y\frac{dy}{dx}-5 = 0 \).

Step2: Solve for \(\frac{dy}{dx}\)

Rearrange the equation \( 2y\frac{dy}{dx}-5 = 0 \) to isolate \(\frac{dy}{dx}\). Add 5 to both sides: \( 2y\frac{dy}{dx}=5 \). Then divide both sides by \( 2y \) (assuming \( y
eq0 \)): \(\frac{dy}{dx}=\frac{5}{2y}\).

Answer:

\(\boldsymbol{\frac{dy}{dx}=\frac{5}{2y}}\) (the option with this expression, e.g., if it's the last option: "The last option: \(\frac{dy}{dx}=\frac{5}{2y}\)")