Question

find the derivative of the function ( y = sqrt{2 - 5x} ).

( \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:

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