question find the derivative of f(x) = (-3x^7 - 3x^6 + 2x^2) / (4x). provide your answer below: f(x) =

Explanation:

Step1: Simplify the function

First, divide each term in the numerator by (4x): [ \begin{align*} f(x)&=\frac{-3x^{7}-3x^{6}+2x^{2}}{4x}\ &=\frac{-3x^{7}}{4x}-\frac{3x^{6}}{4x}+\frac{2x^{2}}{4x}\ &=-\frac{3}{4}x^{6}-\frac{3}{4}x^{5}+\frac{1}{2}x \end{align*} ]

Step2: Apply the power - rule for differentiation

The power - rule states that if (y = ax^{n}), then (y'=nax^{n - 1}). For (y =-\frac{3}{4}x^{6}), (y'=6\times(-\frac{3}{4})x^{6 - 1}=-\frac{9}{2}x^{5}) For (y =-\frac{3}{4}x^{5}), (y'=5\times(-\frac{3}{4})x^{5 - 1}=-\frac{15}{4}x^{4}) For (y=\frac{1}{2}x), (y'=\frac{1}{2}\times1\times x^{1 - 1}=\frac{1}{2})

Step3: Find the derivative of (f(x))

[ \begin{align*} f'(x)&=-\frac{9}{2}x^{5}-\frac{15}{4}x^{4}+\frac{1}{2} \end{align*} ]

Answer:

(-\frac{9}{2}x^{5}-\frac{15}{4}x^{4}+\frac{1}{2})