Advertisements
Advertisements
प्रश्न
If `ysqrt(1 - x^2) + xsqrt(1 - y^2)` = 1, then prove that `(dy)/(dx) = - sqrt((1 - y^2)/(1 - x^2))`
Advertisements
उत्तर
`ysqrt(1 - x^2) + xsqrt(1 - y^2)` = 1
Let sin–1x = A and sin–1y = B.
Then x = sinA and y = sinB
`ysqrt(1 - x^2) + xsqrt(1 - y^2)` = 1
⇒ sinBcosA + sinAcosB = 1
⇒ sin(A + B) = 1
⇒ A + B = sin–11 = `pi/2`
⇒ sin–1x + sin–1y = `pi/2`
Differentiating w.r.to x, we obtain `(dy)/(dx) = - sqrt((1 - y^2)/(1 - x^2))`
APPEARS IN
संबंधित प्रश्न
Differentiate the function with respect to x:
`x^(x^2 -3) + (x -3)^(x^2)`, for x > 3
If (x – a)2 + (y – b)2 = c2, for some c > 0, prove that `[1+ (dy/dx)^2]^(3/2)/((d^2y)/dx^2)` is a constant independent of a and b.
Does there exist a function which is continuos everywhere but not differentiable at exactly two points? Justify your answer?
If sin y = xsin(a + y) prove that `(dy)/(dx) = sin^2(a + y)/sin a`
Let f(x)= |cosx|. Then, ______.
Differential coefficient of sec (tan–1x) w.r.t. x is ______.
| COLUMN-I | COLUMN-II |
| (A) If a function f(x) = `{((sin3x)/x, "if" x = 0),("k"/2",", "if" x = 0):}` is continuous at x = 0, then k is equal to |
(a) |x| |
| (B) Every continuous function is differentiable | (b) True |
| (C) An example of a function which is continuous everywhere but not differentiable at exactly one point |
(c) 6 |
| (D) The identity function i.e. f (x) = x ∀ ∈x R is a continuous function |
(d) False |
cos |x| is differentiable everywhere.
sinx2 + sin2x + sin2(x2)
(x + 1)2(x + 2)3(x + 3)4
`tan^-1 (secx + tanx), - pi/2 < x < pi/2`
`sec^-1 (1/(4x^3 - 3x)), 0 < x < 1/sqrt(2)`
If `"f"("x") = ("sin" ("e"^("x"-2) - 1))/("log" ("x" - 1)), "x" ne 2 and "f" ("x") = "k"` for x = 2, then value of k for which f is continuous is ____________.
If `y = (x + sqrt(1 + x^2))^n`, then `(1 + x^2) (d^2y)/(dx^2) + x (dy)/(dx)` is
A particle is moving on a line, where its position S in meters is a function of time t in seconds given by S = t3 + at2 + bt + c where a, b, c are constant. It is known that at t = 1 seconds, the position of the particle is given by S = 7 m. Velocity is 7 m/s and acceleration is 12 m/s2. The values of a, b, c are ______.
Let S = {t ∈ R : f(x) = |x – π| (e|x| – 1)sin |x| is not differentiable at t}. Then the set S is equal to ______.
If f(x) = `{{:(ax + b; 0 < x ≤ 1),(2x^2 - x; 1 < x < 2):}` is a differentiable function in (0, 2), then find the values of a and b.
The function f(x) = x | x |, x ∈ R is differentiable ______.
For \[v\ne0\], what is the derivative of \[\frac{u}{v}\]?
What is \[\frac{d}{dx}(\sin x)\]?
A function is differentiable at \[x=c\] when which condition holds?
When is a function differentiable on an open interval \[(a,b)\]?
If \[f\] is differentiable at \[c\], which limit equals \[f'(c)\]?
For \[f(x)=|x|\], what is the left-hand derivative at \[x=0\]?
When does a derivative exist?
