Question 3: Find the derivative of the function y = (3x+2) + e2x.

Here is the detailed breakdown for question 3. Question 3: Find the derivative of the function y = (3x+2) + e^2x. This function is a sum of two terms, so we can differentiate each term separately and then add the results. We will use the chain rule for both terms. The chain rule states that if y = f(g(x)), then (dy)/(dx) = f'(g(x)) · g'(x). Step 1: Differentiate the first term, (3x+2). Let u = 3x+2. Then the term becomes (u). The derivative of (u) with respect to u is (u). So, f'(u) = (u). The derivative of u = 3x+2 with respect to x is (du)/(dx) = 3. Applying the chain rule: (d)/(dx)((3x+2)) = (3x+2) · (d)/(dx)(3x+2) (d)/(dx)((3x+2)) = (3x+2) · 3 (d)/(dx)((3x+2)) = 3(3x+2) Step 2: Differentiate the second term, e^2x. Let v = 2x. Then the term becomes e^v. The derivative of e^v with respect to v is e^v. So, f'(v) = e^v. The derivative of v = 2x with respect to x is (dv)/(dx) = 2. Applying the chain rule: (d)/(dx)(e^2x) = e^2x · (d)/(dx)(2x) (d)/(dx)(e^2x) = e^2x · 2 (d)/(dx)(e^2x) = 2e^2x Step 3: Combine the derivatives of the two terms. Since y = (3x+2) + e^2x, the total derivative (dy)/(dx) is the sum of the derivatives found in Step 1 and Step 2. (dy)/(dx) = (d)/(dx)((3x+2)) + (d)/(dx)(e^2x) (dy)/(dx) = 3(3x+2) + 2e^2x The derivative is 3(3x+2) + 2e^2x. Last free one today — make it count tomorrow, or type /upgrade for unlimited.