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.
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Here are the derivatives for each part: a) Find the derivative of y = _a (3x + 2) Step 1: Use the chain rule. Let u = 3x + 2. Then y = _a u. Step 2: Differentiate y with respect to u and u with respect to x. The derivative of _a u is (dy)/(du) = (1)/(u) _a e. The derivative of u = 3x + 2 is (du)/(dx) = 3. Step 3: Apply the chain rule formula (dy)/(dx) = (dy)/(du) × (du)/(dx). (dy)/(dx) = ((1)/(u) _a e) × 3 Substitute u = 3x + 2 back into the expression: (dy)/(dx) = (1)/(3x + 2) × 3 _a e (dy)/(dx) = (3)/(3x + 2) _a e b) Find the derivative of y = _e (4x - 1)^2 Step 1: Use logarithm properties to simplify the expression. y = _e (4x - 1)^2 = 2 _e (4x - 1) Let u = 4x - 1. Then y = 2 _e u. Step 2: Differentiate y with respect to u and u with respect to x. The derivative of 2 _e u (or 2 u) is (dy)/(du) = 2 × (1)/(u) = (2)/(u). The derivative of u = 4x - 1 is (du)/(dx) = 4. Step 3: Apply the chain rule formula (dy)/(dx) = (dy)/(du) × (du)/(dx). (dy)/(dx) = ((2)/(u)) × 4 Substitute u = 4x - 1 back into the expression: (dy)/(dx) = (2)/(4x - 1) × 4 (dy)/(dx) = (8)/(4x - 1) c) Find the derivative of y = _a ((1 - x)/(1 + x)) Step 1: Use logarithm properties to simplify the expression. y = _a (1 - x) - _a (1 + x) Step 2: Differentiate each term using the chain rule. For the first term, (d)/(dx) (_a (1 - x)): Let u = 1 - x, so (du)/(dx) = -1. (d)/(dx) (_a (1 - x)) = (1)/(1 - x) _a e × (-1) = -(1)/(1 - x) _a e For the second term, (d)/(dx) (_a (1 + x)): Let v = 1 + x, so (dv)/(dx) = 1. (d)/(dx) (_a (1 + x)) = (1)/(1 + x) _a e × (1) = (1)/(1 + x) _a e Step 3: Subtract the derivatives. (dy)/(dx) = -(1)/(1 - x) _a e - (1)/(1 + x) _a e Factor out _a e: (dy)/(dx) = _a e (-(1)/(1 - x) - (1)/(1 + x)) Combine the fractions: (dy)/(dx) = _a e ((-(1 + x) - (1 - x))/((1 - x)(1 + x))) (dy)/(dx) = _a e ((-1 - x - 1 + x)/(1 - x^2)) (dy)/(dx) = _a e ((-2)/(1 - x^2)) (dy)/(dx) = -(2)/(1 - x^2) _a e d) Find the derivative of y = _e sqrt(1 + x) Step 1: Use logarithm properties to simplify the expression. y = _e (1 + x)^1/2 = (1)/(2) _e (1 + x) Let u = 1 + x. Then y = (1)/(2) _e u. Step 2: Differentiate y with respect to u and u with respect to x. The derivative of (1)/(2) _e u (or (1)/(2) u) is (dy)/(du) = (1)/(2) × (1)/(u) = (1)/(2u). The derivative of u = 1 + x is (du)/(dx) = 1. Step 3: Apply the chain rule formula (dy)/(dx) = (dy)/(du) × (du)/(dx). (dy)/(dx) = ((1)/(2u)) × 1 Substitute u = 1 + x back into the expression: (dy)/(dx) = (1)/(2(1 + x)) (dy)/(dx) = (1)/(2(1 + x)) That's 3 down. 2 left today — send the next one.