Question

If $y = \sqrt{4-8x}$ then value of $y \cdot \frac{dy}{dx}$ is

• A. -4
• B. 4
• C. 8
• D. -8

Answer 1

Answer: (A)

-4

Answer 2

To find the value of $y \cdot \frac{dy}{dx}$, we are given the function $y = \sqrt{4-8x}$.

Method 1: Direct Differentiation

1. Find $\frac{dy}{dx}$:

Given $y = \sqrt{4-8x}$. We can write this as $y = (4-8x)^{1/2}$. Using the chain rule, $\frac{dy}{dx} = \frac{1}{2}(4-8x)^{(1/2)-1} \cdot \frac{d}{dx}(4-8x)$.

$\frac{dy}{dx} = \frac{1}{2}(4-8x)^{-1/2} \cdot (-8)$.

$\frac{dy}{dx} = \frac{1}{2\sqrt{4-8x}} \cdot (-8)$.

$\frac{dy}{dx} = \frac{-8}{2\sqrt{4-8x}}$.

$\frac{dy}{dx} = \frac{-4}{\sqrt{4-8x}}$.

2. Calculate $y \cdot \frac{dy}{dx}$:

Substitute the expressions for $y$ and $\frac{dy}{dx}$:

$y \cdot \frac{dy}{dx} = (\sqrt{4-8x}) \cdot \left(\frac{-4}{\sqrt{4-8x}}\right)$.

The term $\sqrt{4-8x}$ cancels out.

$y \cdot \frac{dy}{dx} = -4$.

Method 2: Implicit Differentiation

1. Square both sides of the equation for $y$:

Given $y = \sqrt{4-8x}$.

Squaring both sides gives $y^2 = 4-8x$.

2. Differentiate both sides with respect to $x$:

Differentiate $y^2$ with respect to $x$ using the chain rule: $\frac{d}{dx}(y^2) = 2y \frac{dy}{dx}$.

Differentiate $4-8x$ with respect to $x$: $\frac{d}{dx}(4-8x) = 0 - 8 = -8$.

So, $2y \frac{dy}{dx} = -8$.

3. Solve for $y \cdot \frac{dy}{dx}$:

Divide both sides by 2:

$y \cdot \frac{dy}{dx} = \frac{-8}{2}$.

$y \cdot \frac{dy}{dx} = -4$.

Both methods yield the same result.