An application of the calculator thought experiment (CTE) Ý tells us that x sin x is a product;
y = (x)(sin x).
Therefore, by the product rule,
dy
dx
= (1)(sin x) + (x)(cos x) = sin x + x cos x
Recall from Section 2 that
y = cosec x =
1
sin x
.
Therefore, by the quotient rule,
dy
dx
=
(0)(sin x) (1)(cos x)
sin2x
(recall that sin2x is just (sin x)2)
=
cos x
sin2x
=
cos x
sin x
.
1
sin x
=
cotan x cosec x.
(from the identities in Section 2)
Notice that we have just obtained the derivative of one of the remaining five trigonometric functions. Four to go...
© Since the given function is a quotient,
dy
dx
=
(2x+1)(sin x) (x2+x)(cos x)
sin2x
,
and let us just leave it like that (there is no easy simplification of the answer).
(d) Here, an application of the CTEÝ tells us that y is the sine of a quantity.
Since
d
dx
sin x = cos x,
the chain rule (press the pearl to go to the topic summary for a quick review) tells us that
d
dx
sin u = cos u
du
dx
so that
d
dx
sin (3x21) =
cos (3x21)
d
dx
(3x21)
= 6x cos(3x21)
So.......ahh........12
Ken
Edit.......see what happens I give you my working out like every good boy should and jtb beats me