Advertisements
Advertisements
प्रश्न
The derivative of cos–1(2x2 – 1) w.r.t. cos–1x is ______.
विकल्प
2
`(-1)/(2sqrt(1 - x^2)`
`2/x`
1 – x2
Advertisements
उत्तर
The derivative of cos–1(2x2 – 1) w.r.t. cos–1x is 2.
Explanation:
Let y = cos–1(2x2 – 1) and t = cos–1x
Differentiating both the functions w.r.t. x
`"dy"/"dx" = "d"/"dx" cos^-1 (2x^2 - 1)` and `"dt"/"dx" = "d"/"dx" cos^-1x`
⇒ `"dy"/"dx" = (-1)/sqrt(1 - (2x^2 - 1)^2) * "d"/"dx" (2x^2 - 1)` and `"dt"/"dx" = (-1)/sqrt(1 - x^2)`
= `(-1.4x)/sqrt(1 - (4x^4 + 1 - 4x^2)` and `"dt"/"dx" = (-1)/sqrt(1 - x^2)`
= `(-4x)/sqrt(1 - 4x^4 - 1 + 4x^2)`
= `(-4x)/sqrt(4x^2 - 4x^4)`
= `(-4x)/(2xsqrt(1 - x^2)`
⇒ `"dy"/"dx" = (-2)/sqrt(1 - x^2)`
Now `"dy"/"dx" = ("dy"/"dx")/("dt"/"dx")`
= `((-2)/sqrt(1 - x^2))/((-1)/sqrt(1 - x^2))`
= 2.
APPEARS IN
संबंधित प्रश्न
If x cos(a+y)= cosy then prove that `dy/dx=(cos^2(a+y)/sina)`
Hence show that `sina(d^2y)/(dx^2)+sin2(a+y)(dy)/dx=0`
Find the second order derivative of the function.
x2 + 3x + 2
Find the second order derivative of the function.
x . cos x
Find the second order derivative of the function.
x3 log x
Find the second order derivative of the function.
log (log x)
Find the second order derivative of the function.
sin (log x)
If ey (x + 1) = 1, show that `(d^2y)/(dx^2) = (dy/dx)^2`.
If y = (tan–1 x)2, show that (x2 + 1)2 y2 + 2x (x2 + 1) y1 = 2
If `x^3y^5 = (x + y)^8` , then show that `(dy)/(dx) = y/x`
Find `("d"^2"y")/"dx"^2`, if y = `"e"^"x"`
Find `("d"^2"y")/"dx"^2`, if y = `"e"^"log x"`
Find `("d"^2"y")/"dx"^2`, if y = log (x).
Find `("d"^2"y")/"dx"^2`, if y = 2at, x = at2
If x2 + 6xy + y2 = 10, then show that `("d"^2y)/("d"x^2) = 80/(3x + y)^3`
`sin xy + x/y` = x2 – y
(x2 + y2)2 = xy
Derivative of cot x° with respect to x is ____________.
If y = `sqrt(ax + b)`, prove that `y((d^2y)/dx^2) + (dy/dx)^2` = 0.
Read the following passage and answer the questions given below:
|
The relation between the height of the plant ('y' in cm) with respect to its exposure to the sunlight is governed by the following equation y = `4x - 1/2 x^2`, where 'x' is the number of days exposed to the sunlight, for x ≤ 3.
|
- Find the rate of growth of the plant with respect to the number of days exposed to the sunlight.
- Does the rate of growth of the plant increase or decrease in the first three days? What will be the height of the plant after 2 days?
`"Find" (d^2y)/(dx^2) "if" y=e^((2x+1))`
Find `(d^2y)/dx^2` if, `y = e^((2x + 1))`
Find `(d^2y)/(dx^2)` if, y = `e^((2x+1))`
Find `(d^2y)/dx^2` if, y = `e^((2x + 1))`
Find `(d^2y)/dx^2` if, `y = e^((2x + 1))`
If y = 3 cos(log x) + 4 sin(log x), show that `x^2 (d^2y)/(dx^2) + x dy/dx + y = 0`
Let \[y=f(x)\]. What is the first derivative of \[y\] with respect to \[x\]?
Which expression defines the second order derivative of \[y\] with respect to \[x\]?
For \[y=\mathrm{A}\sin x+\mathrm{B}\cos x\], what is \[\frac{d^2y}{dx^2}\]?
If \[y=\sin^{-1}x\], what is \[\frac{dy}{dx}\]?
Which equation is equivalent to \[\frac{dy}{dx}=\frac{1}{\sqrt{1-x^2}}\] for \[y=\sin^{-1}x\]?
After differentiating \[\sqrt{1-x^2}\cdot\frac{dy}{dx}=1\], which equation results?
What is \[\frac{d}{dx}\left(\sqrt{1-x^2}\right)\] in the differentiation for \[y=\sin^{-1}x\]?
Which equation follows from \[(1-x^2)\cdot2y_{1}y_{2}+y_{1}^{2}(0-2x)=0\]?
