What are the differentials for the given functions and rules, including product rule and basic functions?

Welcome back Ahmad — been a while! Let's pick up where you left off. 1. If y = x^2 then dy is: Step 1: Differentiate y with respect to x. (dy)/(dx) = (d)/(dx)(x^2) = 2x Step 2: Express the differential dy. dy = 2x \, dx The correct option is (B) 2xdx. 2. If V = x^3, then differential of V is: Step 1: Differentiate V with respect to x. (dV)/(dx) = (d)/(dx)(x^3) = 3x^2 Step 2: Express the differential dV. dV = 3x^2 \, dx The correct option is (A) 3x^2 dx. 3. d(xy) = Step 1: Apply the product rule for differentials, d(uv) = u \, dv + v \, du. Let u = x and v = y. Then du = dx and dv = dy. Step 2: Substitute into the product rule. d(xy) = x \, dy + y \, dx The correct differential is x \, dy + y \, dx. 4. Differential of sqrt(y) is: Step 1: Let u = sqrt(y) = y^1/2. Step 2: Differentiate u with respect to y. (du)/(dy) = (d)/(dy)(y^1/2) = (1)/(2) y^(1/2 - 1) = (1)/(2) y^-1/2 = (1)/(2sqrt(y)) Step 3: Express the differential du. du = (1)/(2sqrt(y)) \, dy The correct differential is (1)/(2sqrt(y)) \, dy. 5. Value of dy, for y = x^2 and x char: This question is incomplete as the full context for the change in x is not provided. 6. Differential of sqrt(x) is: Step 1: Let y = sqrt(x) = x^1/2. Step 2: Differentiate y with respect to x. (dy)/(dx) = (d)/(dx)(x^1/2) = (1)/(2) x^(1/2 - 1) = (1)/(2) x^-1/2 = (1)/(2sqrt(x)) Step 3: Express the differential dy. dy = (1)/(2sqrt(x)) \, dx The correct differential is (1)/(2sqrt(x)) \, dx. 7. ( 2x)/(4 x) dx = Step 1: Use the double angle identity 2x = 2 x x. (2 x x)/(4 x) dx Step 2: Simplify the integrand. (2 x)/(4) dx = (1)/(2) x \, dx Step 3: Integrate. (1)/(2) x \, dx = (1)/(2) x + C The correct integral is (1)/(2) x + C. 8. ()/(4) dx Step 1: Evaluate the constant ()/(4). ()/(4) = 1 Step 2: Substitute the value into the integral. 1 \, dx Step 3: Integrate. 1 \, dx = x + C The correct integral is x + C. Send me the next one 📸