![If y = [1/3 x3 + 2x + 5]2, find dy/dx. (b) Given that log3 = 0.4771 and log2 = 0.3010, evaluate log162 + log36 - log54.](/_next/image?url=https%3A%2F%2Fohijasnjvprzodcmxltt.supabase.co%2Fstorage%2Fv1%2Fobject%2Fpublic%2Fwhatsapp-images%2F1782855928998-413f92ad44d50fa2.png&w=3840&q=75)
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
a) If y=[(1)/(3)x^3 + 2x + 5]^2, find (dy)/(dx). Step 1: Apply the chain rule. Let u = (1)/(3)x^3 + 2x + 5. Then y = u^2. The derivative of y with respect to u is (dy)/(du) = 2u. Step 2: Find the derivative of u with respect to x. (du)/(dx) = (d)/(dx)((1)/(3)x^3 + 2x + 5) (du)/(dx) = (1)/(3)(3x^3-1) + 2(1) + 0 (du)/(dx) = x^2 + 2 Step 3: Use the chain rule formula (dy)/(dx) = (dy)/(du) · (du)/(dx). (dy)/(dx) = (2u)(x^2 + 2) Substitute u = (1)/(3)x^3 + 2x + 5 back into the expression. (dy)/(dx) = 2((1)/(3)x^3 + 2x + 5)(x^2 + 2) The derivative is: 2((1)/(3)x^3 + 2x + 5)(x^2 + 2) b) Given that 3 = 0.4771 and 2 = 0.3010, evaluate 162 + 36 - 54. Step 1: Use the logarithm properties A + B = (AB) and A - B = ((A)/(B)) to combine the terms. 162 + 36 - 54 = ((162 × 36)/(54)) Step 2: Simplify the expression inside the logarithm. (162 × 36)/(54) = (162)/(54) × 36 Since 162 = 3 × 54: = 3 × 36 = 108 So the expression becomes 108. Step 3: Express 108 as a product of powers of 2 and 3. 108 = 2 × 54 = 2 × 2 × 27 = 2^2 × 3^3 Step 4: Apply logarithm properties (AB) = A + B and (A^n) = n A. 108 = (2^2 × 3^3) = (2^2) + (3^3) = 2 2 + 3 3 Step 5: Substitute the given values for 2 and 3. = 2(0.3010) + 3(0.4771) = 0.6020 + 1.4313 = 2.0333 The evaluated expression is: 2.0333 Got more? Send 'em.