The given function is \( \displaystyle y = \frac{1}{4 \sqrt{4x^{3} – 1}} + \cos^{2}(5x + 8) \) and we need to find the derivative of this function with respect to \( x \).
Solution:
Step 1: Differentiate the first term \( \displaystyle \frac{1}{4 \sqrt{4x^{3} – 1}} \)
\( \displaystyle \frac{d}{dx} \left( \frac{1}{4 \sqrt{4x^{3} – 1}} \right) \\ \displaystyle = \frac{d}{dx} \left( \frac{1}{4} (4x^{3} – 1)^{-1/2} \right) \\ \displaystyle = -\frac{1}{4} \cdot \frac{1}{2} (4x^{3} – 1)^{-3/2} \cdot \frac{d}{dx} (4x^{3} – 1) \\ \displaystyle = -\frac{1}{8} (4x^{3} – 1)^{-3/2} \cdot 12x^{2} \\ \displaystyle = -\frac{12x^{2}}{8} (4x^{3} – 1)^{-3/2} \\ \displaystyle = -\frac{3x^{2}}{2 (4x^{3} – 1)^{3/2}} \\ \)Step 2: Differentiate the Second Term \( \cos^{2}(5x + 8) \)
\( \displaystyle \frac{d}{dx} (\cos^{2}(5x + 8)) \\ \displaystyle = 2 \cos(5x + 8) \cdot (-\sin(5x + 8)) \cdot \frac{d(5x + 8)}{dx} \\ \displaystyle = 2 \cos(5x + 8) \cdot (-\sin(5x + 8)) \cdot 5 \\ \displaystyle = -10 \cos(5x + 8) \sin(5x + 8) \\ \displaystyle = -5 \cdot 2 \sin(5x + 8) \cos(5x + 8) \\ \displaystyle = -5 \sin(2(5x + 8)) \text{ , because } \sin(2a) = 2 \sin(a) \cos(a) \\ \displaystyle = -5 \sin(10x + 16) \\ \)Step 3: Combine the Derivatives
\(\displaystyle \boxed{ \frac{dy}{dx} = -\frac{3x^{2}}{2 (4x^{3} – 1)^{3/2}} – 5 \sin(10x + 16) } \)
Please let me know in the comments if you find any error in this solution.