Advertisements
Advertisements
प्रश्न
If y = cos (sin x), show that: `("d"^2"y")/("dx"^2) + "tan x" "dy"/"dx" + "y" "cos"^2"x" = 0`
उत्तर
y = cos (sin x) ....(i)
`"dy"/"dx" = -"sin" ("sin x") "cos x"` ...(ii)
`("d"^2"y")/("dx"^2)` = sin (sin x) sin x - cos x (cos (sin x)) cos x
= sin (sin x) sin x - cos2x.y .....(using (i))
= sin (sin x) cos x `((-"sin x")/"cos x")` - y cos2x
=`"dy"/"dx"` (- tan x) - y cos2x ......(using(ii))
`("d"^2"y")/("dx"^2) + "tan x" "dy"/"dx" + "y" "cos"^2"x" = 0`
APPEARS IN
संबंधित प्रश्न
If `y=sin^-1(3x)+sec^-1(1/(3x)), ` find dy/dx
Differentiate `tan^(-1)(sqrt(1-x^2)/x)` with respect to `cos^(-1)(2xsqrt(1-x^2))` ,when `x!=0`
Find : ` d/dx cos^−1 ((x−x^(−1))/(x+x^(−1)))`
if `y = sin^(-1)[(6x-4sqrt(1-4x^2))/5]` Find `dy/dx `.
If `y=tan^(−1) ((sqrt(1+x^2)+sqrt(1−x^2))/(sqrt(1+x^2)−sqrt(1−x^2)))` , x2≤1, then find dy/dx.
Find `dy/dx` in the following:
`y = cos^(-1) ((1-x^2)/(1+x^2)), 0 < x < 1`
Find `dy/dx` in the following:
`y = sin^(-1) ((1-x^2)/(1+x^2)), 0 < x < 1`
Find `dx/dy` in the following:
`y = cos^(-1) ((2x)/(1+x^2)), -1 < x < 1`
Find `dy/dx` in the following:
`y = sec^(-1) (1/(2x^2 - 1)), 0 < x < 1/sqrt2`
Differentiate w.r.t. x the function:
`cot^(-1) [(sqrt(1+sinx) + sqrt(1-sinx))/(sqrt(1+sinx) - sqrt(1-sinx))]`, ` 0 < x < pi/2`
Find `dy/dx, if y = sin^-1 x + sin^-1 sqrt (1 - x^2) , 0<x <1`
Differentiate `tan^(-1) ((1+cosx)/(sin x))` with respect to x
if `x = tan(1/a log y)`, prove that `(1+x^2) (d^2y)/(dx^2) + (2x + a) (dy)/(dx) = 0`
Find \[\frac{dy}{dx}\] at \[t = \frac{2\pi}{3}\] when x = 10 (t – sin t) and y = 12 (1 – cos t).
If `log (x^2 + y^2) = 2 tan^-1 (y/x)`, show that `(dy)/(dx) = (x + y)/(x - y)`
Trigonometric and inverse-trigonometric functions are differentiable in their respective domain.
`lim_("h" -> 0) (1/("h"^2 sqrt(8 + "h")) - 1/(2"h"))` is equal to ____________.
`lim_("x"-> 0) ("cosec x - cot x")/"x"` is equal to ____________.
`"d"/"dx" {"cosec"^-1 ((1 + "x"^2)/(2"x"))}` is equal to ____________.
If `"y = sin"^-1 ((sqrt"x" - 1)/(sqrt"x" + 1)) + "sec"^-1 ((sqrt"x" + 1)/(sqrt"x" - 1)), "x" > 0, "then" "dy"/"dx"` is ____________.
Let f(x) = `cos(2tan^-1sin(cot^-1sqrt((1 - x)/x))), 0 < x < 1`. Then ______.
Differentiate `sec^-1 (1/sqrt(1 - x^2))` w.r.t. `sin^-1 (2xsqrt(1 - x^2))`.
