Advertisements
Advertisements
प्रश्न
If f(x) = x3 + x − 2, find (f−1)'(0).
Advertisements
उत्तर
f(x) = x3 + x – 2 ...(1)
Differentiating w.r.t. x, we get
f'(x) = `"d"/"dx"(x^3 + x – 2)`
= 3x2 + 1 - 0
= 3x2 + 1
We know that
`(f^-1) ^' (y) = (1)/(f"'"(x)` ...(2)
From (1), y = f(x) = 0, when x = 0
∴ from (2), (f-1)'(0)
= `(1)/(f"'"(0)) = 1/(3x^2+1)`
`(1)/(f"'"(0)) = 1/(3(0)^2 + 1)` ....[∵ x = 0]
`(1)/(f"'"(0)) = 1/(0 + 1)`
`(1)/(f"'"(0)) = 1/1`
`(1)/(f"'"(0)) = 1`
∴ f'(0) = 1
APPEARS IN
संबंधित प्रश्न
Find the derivative of the function y = f(x) using the derivative of the inverse function x = f–1(y) in the following:
y = `sqrt(x)`
Find the derivative of the function y = f(x) using the derivative of the inverse function x = f-1(y) in the following: y = `sqrt(2 - sqrt(x)`
Find the derivative of the function y = f(x) using the derivative of the inverse function x = f-1(y) in the following: y = `root(3)(x - 2)`
Find the derivative of the function y = f(x) using the derivative of the inverse function x = f–1(y) in the following: y = e2x-3
Find the derivative of the inverse function of the following : y = x2·ex
Find the derivative of the inverse function of the following : y = x cos x
Find the derivative of the inverse function of the following : y = x2 + log x
Find the derivative of the inverse function of the following : y = x log x
Find the derivative of the inverse of the following functions, and also find their value at the points indicated against them. y = x5 + 2x3 + 3x, at x = 1
Find the derivative of the inverse of the following functions, and also find their value at the points indicated against them. y = 3x2 + 2logx3
Using derivative, prove that: sec–1x + cosec–1x = `pi/(2)` ...[for |x| ≥ 1]
Choose the correct option from the given alternatives :
If g is the inverse of function f and f'(x) = `(1)/(1 + x)`, then the value of g'(x) is equal to :
Find the rate of change of demand (x) of a commodity with respect to its price (y) if y = 25x + log(1 + x2)
Find the marginal demand of a commodity where demand is x and price is y.
y = `("x + 2")/("x"^2 + 1)`
Find the marginal demand of a commodity where demand is x and price is y.
y = `(5x + 9)/(2x - 10)`
State whether the following is True or False:
If f′ is the derivative of f, then the derivative of the inverse of f is the inverse of f′.
If y = `"x"^3 + 3"xy"^2 + 3"x"^2"y"` Find `"dy"/"dx"`
Find the derivative of the inverse of function y = 2x3 – 6x and calculate its value at x = −2
Differentiate `tan^-1[(sqrt(1 + x^2) - 1)/x]` w.r. to `tan^-1[(2x sqrt(1 - x^2))/(1 - 2x^2)]`
Choose the correct alternative:
What is the rate of change of demand (x) of a commodity with respect to its price (y) if y = 10 + x + 25x3.
Choose the correct alternative:
If xm. yn = `("x" + "y")^(("m" + "n"))`, then `("dy")/("dx")` = ?
Choose the correct alternative:
If x = at2, y = 2at, then `("d"^2y)/("d"x^2)` = ?
The rate of change of demand (x) of a commodity with respect to its price (y) is ______ if y = 5 + x2e–x + 2x
The rate of change of demand (x) of a commodity with respect to its price (y) is ______ if y = xe–x + 7
State whether the following statement is True or False:
If y = 10x + 1, then `("d"y)/("d"x)` = 10x.log10
State whether the following statement is True or False:
If y = 7x + 1, then the rate of change of demand (x) of a commodity with respect to its price (y) is 7
If `int (dx)/(4x^2 - 1)` = A log `((2x - 1)/(2x + 1))` + c, then A = ______.
I.F. of dx = y (x + y ) dy is a function of ______.
The I.F. of differential equation `dy/dx+y/x=x^2-3 "is" log x.`
Find `dy/dx`, if y = `sec^-1((1 + x^2)/(1 - x^2))`.
If y = `cos^-1 sqrt((1 + x^2)/2`, then `dy/dx` = ______.
If y = `sin^-1((2tanx)/(1 + tan^2x))`, find `dy/dx`.
Find the rate of change of demand (x) of a commodity with respect to its price (y) if y = 12 + 10x + 25x2.
Find the rate of change of demand (x) of a commodity with respect to its price (y) if y = 12 + 10x + 25x2
