find dy for the given values of x and δx. y = √(8x + 3); x = 5; δx = 0.2 what is the derivative of y = √(8x…

Explanation:

Step1: Rewrite the function

Rewrite $y = \sqrt{8x + 3}$ as $y=(8x + 3)^{\frac{1}{2}}$.

Step2: Apply the chain - rule

The chain - rule states that if $y = u^{\frac{1}{2}}$ and $u = 8x+3$, then $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. First, find $\frac{dy}{du}$: $\frac{dy}{du}=\frac{1}{2}u^{-\frac{1}{2}}$. Second, find $\frac{du}{dx}$: $\frac{du}{dx}=8$. Then $\frac{dy}{dx}=\frac{1}{2}(8x + 3)^{-\frac{1}{2}}\cdot8=\frac{4}{\sqrt{8x + 3}}$.

Step3: Calculate $dy$

We know that $dy=\frac{dy}{dx}\cdot\Delta x$. Substitute $x = 5$ into $\frac{dy}{dx}$: $\frac{dy}{dx}\big|_{x = 5}=\frac{4}{\sqrt{8\times5+3}}=\frac{4}{\sqrt{43}}$. Then, since $\Delta x=0.2$, $dy=\frac{4}{\sqrt{43}}\times0.2=\frac{0.8}{\sqrt{43}}\approx0.121$.

Answer:

$\frac{dy}{dx}=\frac{4}{\sqrt{8x + 3}}$; $dy\approx0.121$