If xlog_e(log_ex) − x² + y² = 4(y > 0), then dxdy at x = e is equal to :
03Easy63×since 2002Q3688
For x > 1, if (2x)^2y = 4e^2x−2y,
then (1 + log_e 2x)² dxdy is equal to :
04Easy63×since 2002Q3657
If ƒ(1) = 1, ƒ'(1) = 3, then the derivative of
ƒ(ƒ(ƒ(x))) + (ƒ(x))^2
at x = 1 is :
05Medium63×since 2002Q3658
Let f(x) = log_e(sin x), (0 < x < π) and g(x) = sin^–1
(e^–x
), (x ≥ 0). If α is a positive real number such that
a = (fog)'(α) and b = (fog)(α), then :
06Easy63×since 2002Q3659
Let f : R → R be defined as f(x)=x3+x−5. If g(x) is a function such that f(g(x))=x,∀′x′∈R, then g'(63) is equal to ________________.
07Medium63×since 2002Q3675
If y=sec(tan−1x), then dxdy at x=1 is equal to :
08Easy63×since 2002Q3689
If cos−1(2y)=loge(5x)5,∣y∣<2, then :
09Medium63×since 2002Q3676
If for x∈(0,41), the derivatives of
tan−1(1−9x36xx) is x.g(x), then g(x) equals
10Easy63×since 2002Q3690
The value of loge2dxd(logcosxcosecx) at x=4π is
11Easy63×since 2002Q3656
If g is the inverse of a function f and f′(x)=1+x51, then g′(x) is equal to:
12Easy63×since 2002Q3668
If e^y
+ xy = e, the ordered pair (dxdy,dx2d2y) at x = 0 is equal to :
13Medium63×since 2002Q3652
If x=2cosec−1 and y=2sec−1t(∣t∣≥1), then dxdy is equal to :
14Medium63×since 2002Q3653
The derivative of tan−1(sinx+cosxsinx−cosx), with respect to 2x
, where (x∈(0,2π)) is :
15Medium63×since 2002Q3654
The derivative of
tan−1(x1+x2−1) with
respect to tan−1(1−2x22x1−x2) at x = 21 is :
16Medium63×since 2002Q3655
Let f:(−1,1)→R be a differentiable function with f(0)=−1 and f′(0)=1. Let g(x)=[f(2f(x)+2)]2. Then g′(0)=
17Medium63×since 2002Q3660
Let f(x)=x5+2ex/4 for all x∈R. Consider a function g(x) such that (g∘f)(x)=x for all x∈R. Then the value of 8g′(2) is :
18Hard63×since 2002Q3661
Suppose for a differentiable function h,h(0)=0,h(1)=1 and h′(0)=h′(1)=2. If g(x)=h(ex)eh(x), then g′(0) is equal to:
19Medium63×since 2002Q3662
If y=(x+1+x2)n, then (1+x2)dx2d2y+xdxdy is
20Medium63×since 2002Q3663
Let f(x) be a polynomial function of second degree. If f(1)=f(−1) and a,b,c are in A.P, then f′(a),f′(b),f′(c) are in