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
f(-x) = -((1+x)/(1-x)) Since f(x) = ((1+x)/(1-x)), we can substitute f(x) into the expression for f(-x): f(-x) = -f(x) Therefore, f(x) is an odd function. Step 2: Find an expression for f'(x). Given f(x) = ((1+x)/(1-x)). Using the logarithm property ((a)/(b)) = (a) - (b): f(x) = (1+x) - (1-x) Differentiate f(x) with respect to x. Recall that (d)/(dx)((u)) = (1)/(u) (du)/(dx). For the first term, (d)/(dx)((1+x)) = (1)/(1+x) · (d)/(dx)(1+x) = (1)/(1+x) · 1 = (1)/(1+x). For the second term, (d)/(dx)((1-x)) = (1)/(1-x) · (d)/(dx)(1-x) = (1)/(1-x) · (-1) = -(1)/(1-x). So, f'(x) is: f'(x) = (1)/(1+x) - (-(1)/(1-x)) f'(x) = (1)/(1+x) + (1)/(1-x) Step 3: Simplify f'(x) into a single fraction. Find a common denominator, which is (1+x)(1-x): f'(x) = (1(1-x))/((1+x)(1-x)) + (1(1+x))/((1-x)(1+x)) f'(x) = (1-x+1+x)/((1+x)(1-x)) f'(x) = (2)/(1-x^2) The expression for f'(x) as a single simplified fraction is (2)/(1-x^2). Send me the next one 📸