Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

56_Differentiation of Functions_[Ch. 2
or
564. Find y' , if
2
Solution. In y = ^- In x + In (1 x) In (1 + x 2
) + 3 In sin x +2 In cos x;
u
1
y
, 21, (1)
y' = _- --ui i
3 x 1 x
2x 1 2smx
, _- i-4-3- cosx-- ,
1-j-x 2
sin x cosx
whence
/
y' = y
^
2
-^
1 2x
-~pj
\
2 + 3cotx--2 tan
.
xj
565. Find y\ if y = (smx) x .
Solution. In y x In sinx; -/' = In sin x + x cot x;
/' = (sin x) x (In sin x + x cot x).
,Ln the following problems find y' after first taking logs of the
function y = f(x):
Sec. 3. The Derivatives of Functions Not Represented Explicitly
1. The derivative of an inverse function. I! a function y=f(x) has a
derivative y'x ^ 0, then the derivative of the inverse function x=/- 1
(t/)
*~7
is