DIFFERENTIATION USING THE QUOTIENT RULE
The following problem requires the use of the quotient rule. The derivative of a function h(x) may be denoted by D. The quotient rule is a formal rule for differentiating problems where one function is divided by another. It follows from the limit definition of derivative and is given by:
\[D\frac{f(x)}{g(x)} = \frac{ g(x) D(f(x))  f(x) Dg(x)}{(g(x))^2}\]
Solution:
\[dy/dx = \frac{(2x + cosx) D(4sinx)  (4sinx) D(2x + cosx)}{(2x + cosx)^2}\]
\[= \frac{(2x + cosx)4cosx  4sinx(x  sinx)}{(2 + cosx)^2}\]
\[= \frac{(8xcosx + 4cos^2x)  (8xsinx  4sinx^2x)}{(2 + cosx)^2}\]
\[ = \frac{8(xcosx  sinx) + 4(cos^2x + sin^2x)}{(2 + cosx)^2}\]
We know that Sin^2x + cos^2x = 1
\[ = \frac{8(xcosx  sinx) + 4}{(2 + cosx)^2}\]
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