16. [-/0.88 Points] DETAILS MY NOTES SCALCET9 3.1.051. Find the first and second derivatives of the function. Check to see that your answers are reasonable by comparing the graphs of \( f \), \( f^{\prime} \), and \( f^{\prime \prime} \). \[ f(x)=4 x-5 x^{5 / 6} \] \( f^{\prime}(x)=\square \) \( f^{\prime \prime}(x)=\square \)

Let \[ f(x)=4x-5x^{\frac{5}{6}}. \] **Step 1. Find the first derivative \( f'(x) \):** Differentiate each term separately. 1. The derivative of \(4x\) is \[ \frac{d}{dx}\,4x=4. \] 2. The derivative of \(-5x^{\frac{5}{6}}\) is computed using the power rule: \[ \frac{d}{dx}\,x^{\alpha}=\alpha x^{\alpha-1}. \] Here, \(\alpha=\frac{5}{6}\). Thus, \[ \frac{d}{dx}\,(-5x^{\frac{5}{6}})=-5\cdot\frac{5}{6}x^{\frac{5}{6}-1}=-\frac{25}{6}x^{-\frac{1}{6}}. \] So, the first derivative is \[ f'(x)=4-\frac{25}{6}x^{-\frac{1}{6}}. \] **Step 2. Find the second derivative \( f''(x) \):** Differentiate \( f'(x)=4-\frac{25}{6}x^{-\frac{1}{6}} \). 1. The derivative of the constant \(4\) is \(0\). 2. For the term \(-\frac{25}{6}x^{-\frac{1}{6}}\), use the power rule again with \(\alpha=-\frac{1}{6}\): \[ \frac{d}{dx}\,x^{-\frac{1}{6}}=-\frac{1}{6}x^{-\frac{1}{6}-1}=-\frac{1}{6}x^{-\frac{7}{6}}. \] Multiplying by \(-\frac{25}{6}\) yields: \[ -\frac{25}{6}\cdot\left(-\frac{1}{6}x^{-\frac{7}{6}}\right)=\frac{25}{36}x^{-\frac{7}{6}}. \] Thus, the second derivative is \[ f''(x)=\frac{25}{36}x^{-\frac{7}{6}}. \] **Final Answers:** \[ f'(x)=4-\frac{25}{6}x^{-\frac{1}{6}}, \] \[ f''(x)=\frac{25}{36}x^{-\frac{7}{6}}. \]