Answer to this question on calculus

Here are the solutions to the calculus questions. 1. Find from first principle the differential coefficient of the function y = 2x^2 with respect to x. Step 1: Write down the definition of the derivative from first principles. (dy)/(dx) = _h 0 (f(x+h) - f(x))/(h) Given f(x) = 2x^2, we find f(x+h): f(x+h) = 2(x+h)^2 = 2(x^2 + 2xh + h^2) = 2x^2 + 4xh + 2h^2 Step 2: Calculate f(x+h) - f(x). f(x+h) - f(x) = (2x^2 + 4xh + 2h^2) - 2x^2 = 4xh + 2h^2 Step 3: Divide by h and simplify. (f(x+h) - f(x))/(h) = (4xh + 2h^2)/(h) = (h(4x + 2h))/(h) = 4x + 2h Step 4: Take the limit as h 0. (dy)/(dx) = _h 0 (4x + 2h) = 4x + 2(0) = 4x The differential coefficient is 4x. 2. Differentiate the function with respect to x, y = (1)/(x). Step 1: Rewrite the function using a negative exponent. y = x^-1 Step 2: Apply the power rule for differentiation, (d)/(dx)(x^n) = nx^n-1. (dy)/(dx) = -1 · x^-1-1 = -1 · x^-2 Step 3: Rewrite the result with a positive exponent. (dy)/(dx) = -(1)/(x^2) The derivative is -(1)/(x^2). 3. Find the derivative of the function y = (3x+2) + e^2x. Step 1: Differentiate the first term, (3x+2), using the chain rule. Let u = 3x+2, so (du)/(dx) = 3. (d)/(dx)((3x+2)) = (3x+2) · (d)/(dx)(3x+2) = (3x+2) · 3 = 3(3x+2) Step 2: Differentiate the second term, e^2x, using the chain rule. Let v = 2x, so (dv)/(dx) = 2. (d)/(dx)(e^2x) = e^2x · (d)/(dx)(2x) = e^2x · 2 = 2e^2x Step 3: Add the derivatives of the two terms. (dy)/(dx) = 3(3x+2) + 2e^2x The derivative is 3(3x+2) + 2e^2x. Send me the next one 📸