Question

please solve question 15 and 17 with solution..

Answer 1

15)
y = logtanx
we know ,
if any function, y = logf(x)
then, differentiation ,
dy/dx = f'(x)/f(x)
use this concept here,

differentiate wrt x
dy/dx = 1/tanx .d(tanx)/dx
= sec²x/tanx = 1/cos²x.sinx/cosx
= 1/sinx.cosx = 2/(2sinx.cosx )
=2/sin2x
=2cosecx .


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17)
y = 1/( a - z)
=>y(a - z) = 1
=>ay - yz = 1
=>ay -1 = yz
z = (ay -1)/y

now , differentiate wrt y
dz/dy = { y.d(ay -1)/dy - (ay -1)dy/dy}/y²
= {ay - ay + 1}/y²
=1/y²

hence,
dz/dy = 1/y²
but y = 1/( a - z)
put this
dz/dy = 1/{1/(a - z)² = (a - z)²
dz/dy = ( a - z)²