If x y + 3x + 5y - 1 = 0 then \; dy/dx is

Correct Answer is : 16(x+5)2\; 16/(x+5)²(x+5)216

Correct Answer is : 16(x+5)2\; 16/(x+5)²(x+5)216

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Differential Calculus

Question : 8 of 69

Marks: +1, -0

x y + 3x + 5y - 1 = 0

\; \frac{dy}{dx}

Solution:

Concept:

Implicit differentiation or explicit differentiation using the quotient rule.

Explanation:

Given

xy + 3x + 5y - 1 = 0

To find

\frac{dy}{dx}

, differentiate both sides with respect to

x

\frac{d}{dx}(xy) + 3 + 5\frac{dy}{dx} - 0 = 0

Using product rule:

x\frac{dy}{dx} + y + 3 + 5\frac{dy}{dx} = 0

Collect

\frac{dy}{dx}

terms:

(x+5)\frac{dy}{dx} + (y+3) = 0

Thus

\frac{dy}{dx} = -\frac{y+3}{x+5}

Now solve for

y

from the original equation:

y(x+5) = 1 - 3x \Rightarrow y = \frac{1-3x}{x+5}

Substitute:

\frac{dy}{dx} = -\frac{\frac{1-3x}{x+5}+3}{x+5} = -\frac{\frac{1-3x+3(x+5)}{x+5}}{x+5} = -\frac{\frac{16}{x+5}}{x+5} = -\frac{16}{(x+5)^2}

Answer:

\frac{dy}{dx} = -\frac{16}{(x+5)^2}

, which corresponds to option A.

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