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
Welcome back Ellison — been a while! Let's pick up where you left off. Here are the differentiations for each function: Step 1: Differentiate y = (1)/(2) + sqrt(x) First, rewrite the function using exponent notation. y = (1)/(2) + x^1/2 Now, differentiate each term. The derivative of a constant is 0, and for x^n, the derivative is nx^n-1. (dy)/(dx) = (d)/(dx)((1)/(2)) + (d)/(dx)(x^1/2) (dy)/(dx) = 0 + (1)/(2)x^(1/2)-1 (dy)/(dx) = (1)/(2)x^-1/2 (dy)/(dx) = (1)/(2sqrt(x)) The derivative is (1)/(2sqrt(x)). Step 2: Differentiate y = x The standard derivative of x is - x x. (dy)/(dx) = - x x The derivative is - x x. Step 3: Differentiate y = 2x x^2 This requires the product rule, (d)/(dx)(uv) = u'v + uv', and the chain rule for x^2. Let u = 2x and v = x^2. First, find the derivatives of u and v: u' = (d)/(dx)(2x) = 2 For v = x^2, let w = x^2. Then (dw)/(dx) = 2x. (dv)/(dx) = (d)/(dw)( w) · (dw)/(dx) = (- w) · (2x) = -2x x^2 Now apply the product rule: (dy)/(dx) = u'v + uv' (dy)/(dx) = (2)( x^2) + (2x)(-2x x^2) (dy)/(dx) = 2 x^2 - 4x^2 x^2 The derivative is 2 x^2 - 4x^2 x^2. Send me the next one 📸