Hi. I need help with these questions (see image) Please show workings.

Question

asked Feb 9, 2022 143k views

1 Answer

Mohammadjh · Answer 1 · 2022-02-12T23:47:17+0000

Answer:

see explanation

Explanation:

Using the chain rule

Given

y = f(g(x)), then

(dy)/(dx) = f'(g(x)) × g'(x) ← chain rule

and the standard derivatives

(d)/(dx) (
log_(a) x ) =
(1)/(xlna) ,
(lnx) =
(1)/(x)

(a)

Given

y =
log_(a)
√((1+x))

(dy)/(dx) =
(1)/(lna√((1+x)) ) ×
(d)/(dx) (
$(1+x)^{(1)/(2) }$

=
(1)/(lna√((1+x)) ) ×
(1)/(2)
$(1+x)^{-(1)/(2) }$ ×
(d)/(dx) (1 + x)

=
(1)/(lna√((1+x)) ) ×
(1)/(2√((1+x)) ) × 1

=
(1)/(2lna(1+x))

=
(1)/((1+x)lna^2)

(b)

Given

y = ln sinx

(dy)/(dx) =
(1)/(sinx) ×
(d)/(dx) (sinx)

=
(1)/(sinx) × cosx

=
(cosx)/(sinx)

= cotx