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
Step 1: Identify the function to differentiate. The problem asks to find the first derivative of (2-3x)^5 with respect to x. Let y = (2-3x)^5. Step 2: Apply the chain rule for differentiation. The chain rule states that if y = f(g(x)), then (dy)/(dx) = f'(g(x)) · g'(x). In this case, let g(x) = 2-3x and f(u) = u^5, where u = g(x). Step 3: Differentiate f(u) with respect to u. If f(u) = u^5, then f'(u) = (d)/(du)(u^5) = 5u^5-1 = 5u^4. Step 4: Differentiate g(x) with respect to x. If g(x) = 2-3x, then g'(x) = (d)/(dx)(2-3x) = (d)/(dx)(2) - (d)/(dx)(3x) = 0 - 3 = -3. Step 5: Multiply the results from Step 3 and Step 4, and substitute u back. (dy)/(dx) = f'(g(x)) · g'(x) (dy)/(dx) = 5(2-3x)^4 · (-3) (dy)/(dx) = -15(2-3x)^4 The first derivative of (2-3x)^5 is -15(2-3x)^4. 3 done, 2 left today. You're making progress.