Let \( u \) be a differentiable function of \( x \). Assume the following result is proved. \[ \frac{d}{\Delta x}[1 u]=v^{x} \frac{U}{U T}, U \pm 0 \] Use this result to find the derivative of the function. \[ r(x)-8\left(\left.\frac{x}{4 x} \cos (4 x)-32 \right\rvert\, x \cos (4 x)\right)|8 \cos (4 x)| \]

We will show that, after a little algebra, the given rather “messy” looking function can be rewritten in a much simpler form. (Any answer which is equivalent to the one below is correct.) In what follows we “read” the function   r(x) – 8 ( (x/(4x) cos(4x) – 32) | x cos(4x)| ) |8 cos(4x)| as meaning that r(x) is defined by   r(x) = 8 · ( (x cos(4x) – 32) / (x cos(4x)) ) · (8 cos(4x)). That is, the function is built from the product of three factors. (Sometimes vertical bars are used for “evaluation” or to indicate numerator/denominator; in our interpretation the “bar” in the middle is simply showing that the difference (x cos(4x) – 32) is “divided by” x cos(4x).) Step 1. Write the function in a simpler form. We have   r(x) = 8 · ( (x cos(4x) – 32)/(x cos(4x)) ) · (8 cos(4x)). Multiply the two constants:   8 · (8 cos(4x)) = 64 cos(4x). Thus,   r(x) = 64 cos(4x) · (x cos(4x) – 32)/(x cos(4x)). Now note that the common factor cos(4x) appears in the numerator and denominator. In particular,   (x cos(4x) – 32)/(x cos(4x)) = (x cos(4x))/(x cos(4x)) – 32/(x cos(4x)) = 1 – 32/(x cos(4x)). Thus   r(x)=64 cos(4x) · [1 – 32/(x cos(4x))]      =64 cos(4x) – 64·32/(x)      =64 cos(4x) – 2048/x. So we have rewritten the function in the short form   r(x) = 64 cos(4x) – 2048/x. Step 2. Differentiate r(x). Differentiate term‐by‐term. (1) For 64 cos(4x): Since d/dx[cos(4x)] = –4 sin(4x) (by the chain rule) we have   d/dx[64 cos(4x)] = 64 · (–4 sin(4x)) = –256 sin(4x). (2) For –2048/x: Recall that d/dx[1/x] = –1/x² so   d/dx[–2048/x] = –2048 · (–1/x²) = 2048/x². Thus, the derivative is   r′(x) = –256 sin(4x) + 2048/x². Step 3. Final answer We conclude that an acceptable answer is   r′(x) = –256 sin(4x) + 2048/x². Additional Remarks The problem statement mentioned that “Let u be a differentiable function of x” and assumed a result for the derivative of a composed expression. In our solution the appearance of the function u is hidden in the steps where we used the standard chain rule. (Any result that follows from the chain rule – and which is essentially the “proved” result mentioned – is acceptable here.) Thus the final answer is   r′(x) = 2048/x² – 256 sin(4x). Any answer equivalent to this is correct.