Home
Class 12
MATHS
If g is inverse of f then prove that f''...

If g is inverse of f then prove that `f''(g(x))=-g''(x)(f'(g(x)))^(3).`

Text Solution

Verified by Experts

Since g is inverse of f,f(g(x))=x
`therefore" "f'(g(x))g'(x)=1" (1)"`
`rArr" "f'(g(x))=(1)/(x'(g))`
`rArr" " f''(g(x))g'(x)=-(g''(x))/(g'(x))^(2)`
`rArr" " f''(g(x))=-(g''(x))/(g'(x))^(3)`
`rArr" "f''(g(x))=-g''(x)(f'(g(x)))^(3)`
Promotional Banner

Topper's Solved these Questions

  • DIFFERENTIATION

    CENGAGE PUBLICATION|Exercise Solved Examples|28 Videos

  • DIFFERENTIATION

    CENGAGE PUBLICATION|Exercise Concept Application 3.1|1 Videos

  • DIFFERENTIAL EQUATIONS

    CENGAGE PUBLICATION|Exercise All Questions|578 Videos

  • DOT PRODUCT

    CENGAGE PUBLICATION|Exercise DPP 2.1|15 Videos

Similar Questions

Explore conceptually related problems

Let f(x) and g(x) be real valued functions such that f(x)g(x)=1, AA x in R."If "f''(x) and g''(x)" exists"AA x in R and f'(x) and g'(x) are never zero, then prove that (f''(x))/(f'(x))-(g''(x))/(g'(x))=(2f'(x))/(f(x))

If g(x) is the inverse function and f'(x) = sin x then prove that g'(x) = cosec [g(x)]

If f(x)=x^(3)+3x+4 and g is the inverse function of f(x), then the value of (d)/(dx)((g(x))/(g(g(x)))) at x = 4 equals

If f: RrarrR and g:RrarrR are two given functions, then prove that 2min.{f(x)-g(x),0}=f(x)-g(x)-|g(x)-f(x)|

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________.

If f(x),g(x)a n dh(x) are three polynomial of degree 2, then prove that phi(x)=|[f(x)g(x)h(x)];[f'(x)g'(x) h '(x)];[f' '(x)g' '(x )h ' '(x)]| is: constant

Let f(x)a n dg(x) be two function having finite nonzero third-order derivatives f'''(x) and g'''(x) for all x in R . If f(x).g(x)=1 for all x in R , then prove that (f''')/(f') - (g''')/(g') = 3((f'')/f - (g'')/g) .

If f(x)=x+tanx and g(x) is the inverse of f(x), then differentiation of g(x) is (a) 1/(1+[g(x)-x]^2) (b) 1/(2-[g(x)+x]^2) (c) 1/(2+[g(x)-x]^2) (d) none of these

If g is the inverse of a function f and f'(x)=(1)/(1+x^(5)) , then g'(x) is equal to-

Using the first principle, prove that: d/(dx)(f(x)g(x))=f(x)d/(dx)(g(x))+g(x)d/(dx)(f(x))