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 solutions to the problems: Question 2: We need to find the derivative of ^-1x + ^-1x with respect to x. Step 1: Recall the inverse trigonometric identity ^-1x + ^-1x = ()/(2). Step 2: Substitute the identity into the expression. (d)/(dx)(^-1x + ^-1x) = (d)/(dx)(()/(2)) Step 3: Differentiate the constant. (d)/(dx)(()/(2)) = 0 The correct option is (b). The final answer is 0. Question 3: We need to find the derivative of x with respect to x. Step 1: Recall the standard derivative of x. (d)/(dx)( x) = x x The correct option is (a). The final answer is x x. Question 4: We need to find the derivative of ^-1sqrt(1+x^2) + ^-1sqrt(1+x^2) with respect to x. Step 1: Recall the inverse trigonometric identity ^-1u + ^-1u = ()/(2). Here, u = sqrt(1+x^2). Step 2: Substitute the identity into the expression. (d)/(dx)(^-1sqrt(1+x^2) + ^-1sqrt(1+x^2)) = (d)/(dx)(()/(2)) Step 3: Differentiate the constant. (d)/(dx)(()/(2)) = 0 The correct option is (d). The final answer is 0. Question 5: We need to find (d)/(d(3x))(2^x). This is a derivative with respect to 3x. Step 1: Let y = 2^x and u = 3x. We want to find (dy)/(du). Step 2: Find (dy)/(dx) and (du)/(dx). (dy)/(dx) = (d)/(dx)(2^x) = 2^x 2 (du)/(dx) = (d)/(dx)(3x) = 3 Step 3: Use the chain rule (dy)/(du) = (dy/dx)/(du/dx). (dy)/(du) = (2^x 2)/(3) The result is (1)/(3) 2^x 2, which does not match options (a), (b), or (c). The correct option is (d). The final answer is (1)/(3) 2^x 2. Question 6: Given y = ^-1[( x)/(1+ x)], find (dy)/(dx). Step 1: Simplify the expression inside the ^-1 function.