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 the problems you've selected: 1.9) Given y = 3 2x Step 1: Apply the chain rule. The derivative of u is ^2 u · (du)/(dx). Here, u = 2x, so (du)/(dx) = 2. (dy)/(dx) = 3 · ^2(2x) · (d)/(dx)(2x) (dy)/(dx) = 3 · ^2(2x) · 2 Step 2: Simplify the expression. (dy)/(dx) = 6 ^2(2x) The final answer is 6 ^2(2x). 1.10) Given y = -8 3x Step 1: Apply the chain rule. The derivative of u is (1)/(u) · (du)/(dx). Here, u = 3x, so (du)/(dx) = 3. (dy)/(dx) = -8 · (1)/(3x) · (d)/(dx)(3x) (dy)/(dx) = -8 · (1)/(3x) · 3 Step 2: Simplify the expression. (dy)/(dx) = -(24)/(3x) (dy)/(dx) = -(8)/(x) The final answer is -(8)/(x). 1.19) Given y = (1)/(3^2) Step 1: Rewrite the expression using exponents. y = (1)/((1-x)^2/3) y = (1-x)^-2/3 Step 2: Apply the chain rule. The derivative of u^n is n u^n-1 · (du)/(dx). Here, u = (1-x), so (du)/(dx) = -1. (dy)/(dx) = -(2)/(3)(1-x)^-2/3 - 1 · (d)/(dx)(1-x) (dy)/(dx) = -(2)/(3)(1-x)^-5/3 · (-1) Step 3: Simplify the expression and rewrite with positive exponents. (dy)/(dx) = (2)/(3)(1-x)^-5/3 (dy)/(dx) = (2)/(3(1-x)^5/3) The final answer is (2)/(3(1-x)^5/3). 1.23) Given y = ^2 x + ^2 x - 2 4x Step 1: Use the trigonometric identity ^2 x + ^2 x = 1 to simplify the expression. y = 1 - 2 4x Step 2: Differentiate each term. The derivative of a constant is 0. For -2 4x, apply the chain rule. The derivative of u is ^2 u · (du)/(dx). Here, u = 4x, so (du)/(dx) = 4. (dy)/(dx) = (d)/(dx)(1) - 2 · ^2(4x) · (d)/(dx)(4x) (dy)/(dx) = 0 - 2 · ^2(4x) · 4 Step 3: Simplify the expression. (dy)/(dx) = -8 ^2(4x) The final answer is -8 ^2(4x). 1.25) Given y = (e^x + e^-x)^2 Step 1: Expand the expression. y = (e^x)^2 + 2(e^x)(e^-x) + (e^-x)^2 y = e^2x + 2e^x-x + e^-2x y = e^2x + 2e^0 + e^-2x y = e^2x + 2 + e^-2x Step 2: Differentiate each term. • For e^2x, use the chain rule: (d)/(dx)(e^kx) = ke^kx. So, (d)/(dx)(e^2x) = 2e^2x. • For 2, the derivative is 0. • For e^-2x, use the chain rule: (d)/(dx)(e^-2x) = -2e^-2x. (dy)/(dx) = 2e^2x + 0 - 2e^-2x Step 3: Simplify the expression. (dy)/(dx) = 2e^2x - 2e^-2x The final answer is 2e^2x - 2e^-2x. Got more? Send 'em.