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
Right Victor, let's go. Here are the solutions for the differentiation problems. Question 1: If u = ^-1(3^2), find (du)/(dx). Assumption: The question likely intended to ask for (du)/(d) since the function u is given in terms of . If is a function of x, then (d)/(dx) would be required. We will proceed by finding (du)/(d). Step 1: Identify the function and apply the chain rule for inverse tangent. The derivative of ^-1(f()) with respect to is (f'())/(1 + (f())^2). Here, f() = 3^2. Step 2: Find the derivative of f() with respect to . f'() = (d)/(d)(3^2) = 6 Step 3: Substitute f() and f'() into the derivative formula. (du)/(d) = (6)/(1 + (3^2)^2) (du)/(d) = (6)/(1 + 9^4) Therefore, assuming the question meant (du)/(d): (du)/(d) = (6)/(1 + 9^4) Question 2: Find (d)/(dx)[^-1((1-x^2)/(1+x^2))]. Step 1: Use a trigonometric substitution to simplify the expression inside the inverse sine function. Let x = . This implies = ^-1x. Substitute x = into the expression: (1-x^2)/(1+x^2) = (1-^2)/(1+^2) Recall the double angle identity for cosine: (2) = (1-^2)/(1+^2). So, the expression becomes ^-1((2)). Step 2: Convert (2) to a sine function using a co-function identity. (2) = (()/(2) - 2) Substitute this back into the inverse sine expression: ^-1((()/(2) - 2)) = ()/(2) - 2 Step 3: Substitute back = ^-1x. The expression to differentiate becomes: ()/(2) - 2^-1x Step 4: Differentiate the simplified expression with respect to x. (d)/(dx)(()/(2) - 2^-1x) = (d)/(dx)(()/(2)) - 2(d)/(dx)(^-1x) The derivative of a constant (()/(2)) is 0. The derivative of ^-1x is (1)/(1+x^2). = 0 - 2((1)/(1+x^2)) = -(2)/(1+x^2) -(2)/(1+x^2) Got more? Send 'em.