Given that f(x) = x ( x), find f'(x) when x = (pi)/(4).

You're on a roll — It seems there was some confusion with the numbering in our previous exchanges, as the image you sent contains questions 1 through 5. I have already solved Question 1 (parts a and b) and Question 4 ii) from this image, which I previously referred to as "Question 8" and "Question 7" respectively. Let's continue by solving Question 3 from the image. --- Question 3: Given that f(x) = x ( x), find f'(x) when x = ()/(4). Step 1: Identify the function and the required operation. We need to find the derivative f'(x) of f(x) = x ( x) using the product rule, and then evaluate it at x = ()/(4). The product rule states that if f(x) = u(x)v(x), then f'(x) = u'(x)v(x) + u(x)v'(x). Step 2: Define u(x) and v(x) and find their derivatives. Let u(x) = x. Then u'(x) = x. Let v(x) = ( x). To find v'(x), we use the chain rule. Let w = x, so v(x) = w. (dv)/(dw) = (1)/(w) = (1)/( x). (dw)/(dx) = ^2 x. So, v'(x) = (dv)/(dw) · (dw)/(dx) = (1)/( x) · ^2 x. We can simplify this: v'(x) = ( x)/( x) · (1)/(^2 x) = (1)/( x x) Step 3: Apply the product rule to find f'(x). f'(x) = u'(x)v(x) + u(x)v'(x) f'(x) = ( x)(( x)) + ( x)((1)/( x x)) f'(x) = x ( x) + (1)/( x) f'(x) = x ( x) + x Step 4: Evaluate f'(x) at x = ()/(4). Substitute x = ()/(4) into the expression for f'(x): f'(()/(4)) = (()/(4)) ((()/(4))) + (()/(4)) Recall the trigonometric values: (()/(4)) = sqrt(2)2 (()/(4)) = 1 (()/(4)) = (1)/(()4) = (1)/(sqrt(2)/2) = (2)/(sqrt(2)) = sqrt(2) Also, (1) = 0. Substitute these values: f'(()/(4)) = (sqrt(2)2) · (0) + sqrt(2) f'(()/(4)) = 0 + sqrt(2) f'(()/(4)) = sqrt(2) The value of f'(x) when x = ()/(4) is sqrt(2). That's 2 down. 3 left today — send the next one.