Let f(x)=x^(3),x in (0,oo) and let g(x) ...

Let `f(x)=x^(3),x in (0,oo)` and let g(x) be inverse of f(x), then g'(x) must be

A

`x(1+log x)`

B

`x(1+ log (x))`

C

`(1)/(x(1+log g(x))`

D

non-existent

Text Solution

Verified by Experts

`f(x)=x^(x),g(x)" is inverse of "f(x)`
`therefore" "f(g(x))=x`
`rArr" "f'(g(x))cdotg'(x)=1`
`rArr" "g'(x)=(1)/(f'(g(x)))`
`"Now "log_(e)f(x)=xlog_(e)x`
`therefore" "(f'(x))/(f(x))=x(1)/(x)+log_(e)x=1+log_(e)x`
`therefore" "f'(x)=x^(x)(1+log_(e)x)`
`therefore" "f'(g(x))=(g(x))^(g(x))(1+log g(x))`
`"Now "f(g(x))=g(x)^(g(x))=x`
`therefore" "f'(g(x))=x(1+log g(x))`
`therefore" "g'(x)=(1)/(x(1+log g(x)))`

Promotional Banner

Promotional Banner

Topper's Solved these Questions

DIFFERENTIATION
CENGAGE ENGLISH|Exercise Multiple Correct Answers Type|29 Videos
DIFFERENTIATION
CENGAGE ENGLISH|Exercise Linked Comprehension Type|24 Videos
DIFFERENTIATION
CENGAGE ENGLISH|Exercise Concept Application 3.9|14 Videos
DIFFERENTIAL EQUATIONS
CENGAGE ENGLISH|Exercise Matrix Match Type|5 Videos
DOT PRODUCT
CENGAGE ENGLISH|Exercise DPP 2.1|15 Videos

Similar Questions

Explore conceptually related problems

f(x)=x^x , x in (0,oo) and let g(x) be inverse of f(x) , then g(x)' must be

if f(x)=x^x,x in (1,infty) and g(x) be inverse function of f(x) then g^(')(x) must be equal to

Let f(x) = x + cos x + 2 and g(x) be the inverse function of f(x), then g'(3) equals to ........ .

Consider a function f(x)=x^(x), AA x in [1, oo) . If g(x) is the inverse function of f(x) , then the value of g'(4) is equal to

Let f(x)=x^(2)-4x-3, x gt2 and g(x) be the inverse of f(x) . Then the value fo (g') , where f(x)=2 , is (here, g' represents the first derivative of g)

Let f(x)=x^(3)+x+1 and let g(x) be its inverse function then equation of the tangent to y=g(x) at x = 3 is

Let f (x) = x^(3)+ 4x ^(2)+ 6x and g (x) be inverse then the vlaue of g' (-4):

If f(x)=x+tanx and f is inverse of g, then g ′(x) is equal to

Let f(x)=int_(0)^(x)(dt)/(sqrt(1+t^(3))) and g(x) be the inverse of f(x) . Then the value of 4 (g''(x))/(g(x)^(2)) is________.

Let f(x)=int_2^x(dt)/(sqrt(1+t^4))a n dg(x) be the inverse of f(x) . Then the value of g'(0)

CENGAGE ENGLISH-DIFFERENTIATION-Execrises

If y=x^((x^(x))) then (dy)/(dx) is
Text Solution
|
The first derivative of the function [cos^(-1)(sinsqrt((1+x)/(2)))+x^(...
Text Solution
|
Let f(x)=x^(3),x in (0,oo) and let g(x) be inverse of f(x), then g'(x)...
Text Solution
|
If y=ax^(n+1)+bx^(-n)," then "x^(2)(d^(2)y)/(dx^(2)) is equal to
Text Solution
|
If y=ax^(n+1)+bx^(-n)," then "x^(2)(d^(2)y)/(dx^(2)) is equal to
Text Solution
|
Suppose f(x)=e^(ax) + e^(bx), where a!=b, and that fprimeprime(x)-2fpr...
Text Solution
|
(d^(20)y)/(dx^(20))(2cosxcos3x)is equal to 2^(20)(cos2x-2^(20)os3x) ...
Text Solution
|
(d^n)/(dx^n)(logx)=? (a)((n-1)!)/(x^n) (b) (n !)/(x^n) (c)((n-2)!)/(...
Text Solution
|
The n t h derivative of the function f(x)=1/(1-x^2)[w h e r ex in (-1,...
Text Solution
|
If y=xlog"{"x/((a+b x))"]" , then show that x^3(d^2y)/(dx^2)=(x(dy)/(d...
Text Solution
|
If a x^2+2h x y+b y^2=1,t h e n(d^(2y))/(dx^2) is (h^2-a b)/((h x+b y)...
Text Solution
|
"If "y^(1//m)=(x+sqrt(1+x^(2)))," then "(1+x^(2))y(2)+xy(1) is (where ...
Text Solution
|
If (sinx)(cosy)=1/2,t h e n(d^2y)/(dx^2)a t(pi/4,pi/4) is -4 (b) -2 (c...
Text Solution
|
A function f satisfies the condition f(x)=f^(prime)(x)+f''(x)+f'''(x)....
Text Solution
|
Let f(x) be a polynomial of degree 3 such that f(3)=1,f^(prime)(3)=-1,...
Text Solution
|
If y^2=a x^2+b x+c , then y^3(d^2y)/(dx^2) is (a) a constant (b) a fun...
Text Solution
|
If y=x+e^x ,t h e n(d^(2x))/(dy^2)i s e^x (b) -(e^x)/((1+e^x)^3) -(e...
Text Solution
|
if y= sin mx, them the value of the determinant |{:(y,,y(1),,y(...
Text Solution
|
If f^(x)=-f(x)a n dg(x)=f^(prime)(x)a n d F(x)=(f(x/2))^2+(g(x/2))^2 ...
Text Solution
|
Let y=1n(1+cosx)^2 . Then the value of (d^2y)/(dx^2)+2/(e^(y/2)) equal...
Text Solution
|