Understanding Differentiation - JEE Main 2014 Maths
1. Differentiation
Differentiation of parametric functions
Sometimes x and y are given as functions of a
single variable, e.g., x = Φ (t), y = ψ (t) are two
functions and t is a variable. In such a case x
and y are called parametric functions or
parametric equations and t is called the
dy
parameter. To find dx in case of parametric
functions, we first obtain the relationship
between x and y by eliminating the parameter t
and then we differentiate it with respect to x.
But every time it is not convenient to eliminate
the parameter.
dy
Therefore
can also be obtained by the
dx
following formula
2. dy dy / dt
dx dx / dt
To prove it, let Δ x and Δ y be the changes in x
and y respectively corresponding to a small
change Δ t in t.
since
y y / t
,
x x / t
y dy
dy
y t0 t
'(t)
lim
dt
dx x 0 x lim x dx '(t)
t 0 t
dt
lim
3. Multiple Choice Questions
1. If
t
x a cos t log tan , y a sin t
2
dy
dx
(a) tan t
(b) -tant
(c) cot t
(d) – cot t
then
4. Ans:
(a) tan t
t
x a cos t log tan
Given that
2
and y a sin t . Differentiating with respect
to t,
dy
a cos t
we get dt
.....(i)
dx
t 1
2 t
a sin t cot sec
and dt
2 2
2
1
cos 2 t
a sin t
a cos t cot t
a
sin t
sin t
.....(ii)
dy
From (ii) and (i), we get dx tan t .
5. d2 y
2
2. If x at , y 2at, then dx 2
(a)
1
t2
1
3
(c) t
1
3
(b) 2at
1
(d) 2at 3
6. Ans:
1
3
(d) 2at
dy dy / dt 2a dy 1 2a
y
dx dx / dt 2at , dx t
2
dy
d 2 y dy
y
2a y 2 0
, dx dx
dx
d 2 y (dy / dx )2
1
2
y
dx
2at 3 .
OR
d2y/dx2 = d/dx (dy/dx) =
[d/dt
(dy/dx)]
/
[dx/dt]
= [d/dt (1/t)] / [2at] = (-1/t2)/2at = 1/2at3
7. 3. If x 2cos t cos 2t , y 2sin t sin 2t ,
dy
then at t 4 , dx
(a)
2 1
(b)
(c)
2 1
2
(d) None of these
2 1
8. Ans:
(a)
2 1
dx
2 sin t 2 sin 2t
dt
dy
2 cos t 2 cos 2t
dt
dy cos t cos 2t
dx
sin 2t sin t
dy
t ,
Put
4 we have dx t / 4
cos / 4 cos / 2
2 1
.
sin / 2 sin / 4
and
9. Multiple Choice Questions
1. If
x a sin 2 1 2cos , y bcos 2 1 cos 2 ,
dy
then dx
b tan
(a) a
a tan
(b) b
a
(c) b tan
b
(d) a tan
dy
3at
3at 2
x
,y
, then
3
3
2. If
dx
1 t
1 t
t 2 t3
(a) 1 2t 3
t 2 t3
(c) 1 2t 3
t 2 t3
(b) 1 2t 3
t 2 t3
(d) 1 2t 3
10. 3. If x = a (t + sin t) and y = a (1 – cos t), then
dy
dx equals
(a) tan(t / 2)
(b) cot(t / 2)
(c) tan 2t
(d) tan t
11. Answers
1.
b tan
(a) a
1
x a sin 2 sin 4
2
,
1
y b cos 2 (1 cos 4 )
2
dx
2a(cos 2 cos 4 ) 2a.2 cos 3 cos
d
and
dy
2b(sin 4 sin 2 ) 2b.2 cos 3 sin
d
dy dy dx b
tan
.
dx d d a
12. 2.
t 2 t3
(b) 1 2t 3
3at
3at 2
x
, y
y tx .
3
1t
1 t 3 . Clearly,
dy
dt
t. 1 x .
Differentiate w.r.t. x, we get dx
dx
…..(i)
Now,
dx
1 t 3 t. 3 t 2 3a.(1 2 t 3 )
3 a.
3 2
dt
(1 t )
(1 t 3 )2
.....(ii)
dy
3 at
(1 t 3 )2
t(2 t 3 )
t
.
3
3
dx
1 t 3 a(1 2 t ) 1 2 t 3 , {by (ii)}.
13. 3.
(a) tan(t / 2)
d
[a(1 cos t)]
dy dy / dt
dt
d
dx dx / dt
[a(t sin t)]
dt
t
t
2 sin cos
dy
a sin t
sin t
2
2
t
dx a a cos t 1 cos t
2 cos 2
2
dy
t
tan
dx
2.
Differentiation of infinite series
If y is given in the form of infinite series of x and
we have to find out
dy
dx
then we remove one or
more terms, it does not affect the series
14. i.
if y f(x) f(x) f(x) ....... ,
theny f(x) y y f(x) y
2
2y
dy
dy
f (x)
dx
dx
,
dy
f (x)
,
dx 2y 1
ii.
f(x).....
If y f(x)f(x)f(x)
logy y logf(x)
then y f(x)y
1 dy y.f (x)
dy
dy
y 2f (x)
logf(x). ,
y dx
f(x)
dx
dx f(x)[1 y logf(x)]
iii.
if y f(x)
1
f(x)
1
f(x) ....
then
dy
yf (x)
dx 2y f(x)
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