Question

differentiate the following w.r.t x
a) y= tan(lnx)
b) y= ln(x^2 + 1)
plz explain every single step such as in part (a) tan will be eleminated but idk the reason....
thank you

Answer 1

(a) y = tan(lnx)
we know, if y = tanf(x)
then, dy/dx = sec²f(x).df(x)/dx
use this concept here,
dy/dx = sec²(lnx). d(lnx)/dx
dy/dx = sec²(lnx) . 1/x
[ sec²∅- tan²∅ = 1 use this ]

dy/dx = {1 + tan²(lnx)}/x
but
tan(lnx) = y
so, dy/dx = (1 + y²)/x

hence, dy/dx = (1 + y²)/x

(b) y = ln(x² + 1)
we know, if y = lnf(x)
dy/dx = 1/f(x) .df(x)/dx use this concept .

y = ln(x² + 1)
dy/dx = 1/(x² +1) .d(x² +1)/dx
dy/dx = 1/(x² +1) .2x

hence, dy/dx = 2x/(x² +1)