find the derivative of the function ( y = sqrt{2 - 5x} ).\n\n( \frac{dy}{dx} = square )

Explanation:

Step1: Rewrite the function

Rewrite (y = \sqrt{2 - 5x}) as (y=(2 - 5x)^{\frac{1}{2}}).

Step2: Apply the chain rule

The chain rule states that if (y = f(g(x))), then (y^\prime=f^\prime(g(x))\cdot g^\prime(x)). Let (u = 2-5x), so (y = u^{\frac{1}{2}}). First, find (\frac{dy}{du}) and (\frac{du}{dx}). (\frac{dy}{du}=\frac{1}{2}u^{-\frac{1}{2}}) (using the power rule (\frac{d}{du}(u^n)=nu^{n - 1})), and (\frac{du}{dx}=- 5).

Step3: Substitute back

By the chain rule (\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}). Substitute (u = 2-5x) into (\frac{dy}{du}): (\frac{dy}{dx}=\frac{1}{2}(2 - 5x)^{-\frac{1}{2}}\cdot(-5)) Simplify the expression: (\frac{dy}{dx}=-\frac{5}{2\sqrt{2 - 5x}})

Answer:

(-\frac{5}{2\sqrt{2 - 5x}})