This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Hey Ellison, good to see you again. Here are the differentiations for Question 5: I. Differentiate y = (1)/(x) + sqrt(x) Step 1: Rewrite the function using exponents. y = x^-1 + x^1/2 Step 2: Apply the power rule for differentiation, (d)/(dx)(x^n) = nx^n-1. (dy)/(dx) = -1 · x^-1-1 + (1)/(2) · x^1/2-1 (dy)/(dx) = -x^-2 + (1)/(2)x^-1/2 Step 3: Rewrite with positive exponents and radicals. (dy)/(dx) = -(1)/(x^2) + (1)/(2sqrt(x)) II. Differentiate y = x Step 1: Recall the standard derivative of x. (dy)/(dx) = - x x III. Differentiate y = 2x x^2 Step 1: Use the product rule (d)/(dx)(uv) = u'v + uv', where u = 2x and v = x^2. Step 2: Find the derivatives of u and v. u = 2x u' = 2 For v = x^2, use the chain rule. Let w = x^2, so v = w. (dv)/(dx) = (dv)/(dw) · (dw)/(dx) = (- w) · (2x) = -2x x^2 So, v' = -2x x^2. Step 3: Apply the product rule. (dy)/(dx) = (2)( x^2) + (2x)(-2x x^2) (dy)/(dx) = 2 x^2 - 4x^2 x^2 (dy)/(dx) = 2 x^2 - 4x^2 x^2 Send me the next one 📸