Question

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find the derivative of the function.
f(x) = √(7x² - 8x + 7)
f(x) =
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Explanation:

Step1: Rewrite the function

Rewrite $f(x)=\sqrt{7x^{2}-8x + 7}=(7x^{2}-8x + 7)^{\frac{1}{2}}$

Step2: Apply the chain - rule

The chain - rule states that if $y = u^{\frac{1}{2}}$ and $u=7x^{2}-8x + 7$, then $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. First, find $\frac{dy}{du}$: $\frac{d}{du}(u^{\frac{1}{2}})=\frac{1}{2}u^{-\frac{1}{2}}$. Then find $\frac{du}{dx}$: $\frac{d}{dx}(7x^{2}-8x + 7)=14x-8$.

Step3: Calculate the derivative

$\frac{dy}{dx}=\frac{1}{2}(7x^{2}-8x + 7)^{-\frac{1}{2}}\cdot(14x - 8)=\frac{14x-8}{2\sqrt{7x^{2}-8x + 7}}=\frac{7x - 4}{\sqrt{7x^{2}-8x + 7}}$

Answer:

$\frac{7x - 4}{\sqrt{7x^{2}-8x + 7}}$