Advertisements
Advertisements
प्रश्न
Using derivative, prove that: sec–1x + cosec–1x = `pi/(2)` ...[for |x| ≥ 1]
उत्तर
Let f(x) = sec–1x + cosec–1x for |x| ≥ 1 ...(1)
Differentiating w.r.t. x, we get
f'(x) = `"d"/"dx"(sec^-1 x + "cosec"^-1 x)`
= `"d"/"dx"(sec^-1 x) + "d"/"dx"("cosec"^-1 x)`
= `(1)/(xsqrt(x^2 - 1)) - (1)/(xsqrt(x^2 - 1)`
= 0
Since, f'(x) = 0, f(x) is a constant function.
Let f(x) = k.
For any value of x,f(x) = k, where |x| > 1
Let x = 2.
Then, f(2) = k ...(2)
From (1),f(2) = sec–1(2) + cosec–1(2)
= `pi/(3) + pi/(6) = pi/(2)`
∴ k = `pi/(2)` ...[By (2)]
∴ f(x) = k = `pi/(2)`
Hence, sec–1x + cosec–1x = `pi/(2)`. ...[By (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 = log(2x – 1)
Find the derivative of the function y = f(x) using the derivative of the inverse function x = f–1(y) in the following: y = 2x + 3
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 = 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 = 3x2 + 2logx3
If f(x) = x3 + x – 2, find (f–1)'(0).
Using derivative, prove that: tan –1x + cot–1x = `pi/(2)`
If y = f(x) is a differentiable function of x, then show that `(d^2x)/(dy^2) = -(dy/dx)^-3.("d^2y)/(dx^2)`.
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 = `(5"x" + 9)/(2"x" - 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"`
If `"x"^3 + "y"^2 + "xy" = 7` Find `"dy"/"dx"`
If y = `tan^-1((2x)/(1 - x^2))`, x ∈ (−1, 1) then `("d"y)/("d"x)` = ______.
If g is the inverse of f and f'(x) = `1/(1 + x^4)` then g'(x) = ______
Find the derivative of the inverse of function y = 2x3 – 6x and calculate its value at x = −2
Let f(x) = x5 + 2x – 3 find (f−1)'(-3)
Find the derivative of cos−1x w.r. to `sqrt(1 - 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 x = at2, y = 2at, then `("d"^2y)/("d"x^2)` = ?
Find rate of change of demand (x) of a commodity with respect to its price (y) if y = `(3x + 7)/(2x^2 + 5)`
The rate of change of demand (x) of a commodity with respect to its price (y), if y = 20 + 15x + x3.
Solution: Let y = 20 + 15x + x3
Diff. w.r.to x, we get
`("d"y)/("d"x) = square + square + square`
∴ `("d"y)/("d"x)` = 15 + 3x2
∴ By derivative of the inverse function,
`("d"x)/("d"y) 1/square, ("d"y)/("d"x) ≠ 0`
∴ Rate of change of demand with respect to price = `1/(square + square)`
If `int (dx)/(4x^2 - 1)` = A log `((2x - 1)/(2x + 1))` + c, then A = ______.
Find the rate of change of demand (x) of a commodity with respect to its price (y) if y = 12 + 10x + 25x2
I.F. of dx = y (x + y ) dy is a function of ______.
If y = `cos^-1 sqrt((1 + x^2)/2`, then `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+25x^2`
