Problem 5: Find the derivative of y = x5 [3]x-1(2x+3)4 using logarithmic differentiation.

Okay, let's re-explain Problem 5, focusing on each step of logarithmic differentiation. Problem 5: Find the derivative of y = x^5 [3]x-1(2x+3)^4 using logarithmic differentiation. Logarithmic differentiation is useful when you have a function that involves products, quotients, and powers, especially when the base and exponent both contain variables. Step 1: Take the natural logarithm of both sides of the equation. This allows us to use logarithm properties to simplify the expression before differentiating. y = x^5 (x-1)^1/3(2x+3)^4 y = (x^5 (x-1)^1/3(2x+3)^4) Step 2: Use logarithm properties to expand the right side. Recall these properties: • (AB) = A + B • ((A)/(B)) = A - B • (A^c) = c A Applying these properties: y = (x^5) + ((x-1)^1/3) - ((2x+3)^4) y = 5 x + (1)/(3) (x-1) - 4 (2x+3) Step 3: Differentiate both sides with respect to x implicitly. Remember that (d)/(dx)( y) = (1)/(y)(dy)/(dx). For the right side, we differentiate each term. (1)/(y) (dy)/(dx) = (d)/(dx)(5 x) + (d)/(dx)((1)/(3) (x-1)) - (d)/(dx)(4 (2x+3)) Differentiating each term: • (d)/(dx)(5 x) = 5 · (1)/(x) = (5)/(x) • (d)/(dx)((1)/(3) (x-1)) = (1)/(3) · (1)/(x-1) · (d)/(dx)(x-1) = (1)/(3(x-1)) · 1 = (1)/(3(x-1)) (using the chain rule) • (d)/(dx)(4 (2x+3)) = 4 · (1)/(2x+3) · (d)/(dx)(2x+3) = (4)/(2x+3) · 2 = (8)/(2x+3) (using the chain rule) Substitute these derivatives back into the equation: (1)/(y) (dy)/(dx) = (5)/(x) + (1)/(3(x-1)) - (8)/(2x+3) Step 4: Solve for (dy)/(dx). Multiply both sides by y: (dy)/(dx) = y ( (5)/(x) + (1)/(3(x-1)) - (8)/(2x+3) ) Finally, substitute the original expression for y back into the equation: (dy)/(dx) = x^5 [3]x-1(2x+3)^4 ( (5)/(x) + (1)/(3(x-1)) - (8)/(2x+3) ) 3 done, 2 left today. You're making progress.