find the derivative of f(x) = 3 / √(4x² - 2.4x - 4π).\n○ f(x) = -3/2 (8x - 2.4)^(-3/2)\n○ f(x) = -3/2 (4x²…

Explanation:

Step1: Rewrite the function

Rewrite $f(x)=\frac{3}{\sqrt{4x^{2}-2.4x - 4\pi}}$ as $f(x)=3(4x^{2}-2.4x - 4\pi)^{-\frac{1}{2}}$.

Step2: Apply the chain - rule

The chain - rule states that if $y = u^n$ and $u = g(x)$, then $\frac{dy}{dx}=n\cdot u^{n - 1}\cdot u'$. Here, $n=-\frac{1}{2}$, $u = 4x^{2}-2.4x - 4\pi$, and $u'=\frac{d}{dx}(4x^{2}-2.4x - 4\pi)=8x - 2.4$. So, $f'(x)=3\times(-\frac{1}{2})(4x^{2}-2.4x - 4\pi)^{-\frac{1}{2}-1}\times(8x - 2.4)$.

Step3: Simplify the expression

$f'(x)=-\frac{3}{2}(4x^{2}-2.4x - 4\pi)^{-\frac{3}{2}}(8x - 2.4)$.

Answer:

$f'(x)=-\frac{3}{2}(4x^{2}-2.4x - 4\pi)^{-\frac{3}{2}}(8x - 2.4)$