Advertisements
Advertisements
Question
Find `"dy"/"dx" if, sqrt"x" + sqrt"y" = sqrt"a"`
Solution
`sqrt"x" + sqrt"y" = sqrt"a"`
Differentiating both sides w.r.t. x, we get
`1/(2 sqrt"x") + 1/(2sqrt"y") * "dy"/"dx" = 0`
∴ `1/(2sqrt"y") * "dy"/"dx" = (-1)/(2sqrt"x")`
∴ `"dy"/"dx" = - sqrt("y"/"x")`
APPEARS IN
RELATED QUESTIONS
If y=eax ,show that `xdy/dx=ylogy`
If xpyq = (x + y)p+q then Prove that `dy/dx = y/x`
Find `dx/dy` in the following.
x2 + xy + y2 = 100
Find `dy/dx` in the following:
`y = sin^(-1)((2x)/(1+x^2))`
Examine the differentialibilty of the function f defined by
\[f\left( x \right) = \begin{cases}2x + 3 & \text { if }- 3 \leq x \leq - 2 \\ \begin{array}xx + 1 \\ x + 2\end{array} & \begin{array} i\text { if } - 2 \leq x < 0 \\\text { if } 0 \leq x \leq 1\end{array}\end{cases}\]
If f (x) = |x − 2| write whether f' (2) exists or not.
If x = tan-1t and y = t3 , find `(dy)/(dx)`.
Discuss extreme values of the function f(x) = x.logx
Find `"dy"/"dx"` if x = a cot θ, y = b cosec θ
Find `"dy"/"dx"` if : x = cosec2θ, y = cot3θ at θ= `pi/(6)`
Find `"dy"/"dx"` if : x = a cos3θ, y = a sin3θ at θ = `pi/(3)`
Find `"dy"/"dx"` if : x = t + 2sin (πt), y = 3t – cos (πt) at t = `(1)/(2)`
Differentiate `tan^-1((sqrt(1 + x^2) - 1)/(x)) w.r.t tan^-1((2xsqrt(1 - x^2))/(1 - 2x^2))`.
If 2y = `sqrt(x + 1) + sqrt(x - 1)`, show that 4(x2 – 1)y2 + 4xy1 – y = 0.
Find the nth derivative of the following : apx+q
Find the nth derivative of the following : `(1)/(3x - 5)`
If y `tan^-1(sqrt((a - x)/(a + x)))`, where – a < x < a, then `"dy"/"dx"` = .........
Suppose that the functions f and g and their derivatives with respect to x have the following values at x = 0 and x = 1:
| x | f(x) | g(x) | f')x) | g'(x) |
| 0 | 1 | 5 | `(1)/(3)` | |
| 1 | 3 | – 4 | `-(1)/(3)` | `-(8)/(3)` |
(i) The derivative of f[g(x)] w.r.t. x at x = 0 is ......
(ii) The derivative of g[f(x)] w.r.t. x at x = 0 is ......
(iii) The value of `["d"/"dx"[x^(10) + f(x)]^(-2)]_(x = 1_` is ........
(iv) The derivative of f[(x + g(x))] w.r.t. x at x = 0 is ...
Differentiate the following w.r.t. x:
`tan^-1(x/(1 + 6x^2)) + cot^-1((1 - 10x^2)/(7x))`
If `sqrt(y + x) + sqrt(y - x)` = c, show that `"dy"/"dx" = y/x - sqrt(y^2/x^2 - 1)`.
If `xsqrt(1 - y^2) + ysqrt(1 - x^2)` = 1, then show that `"dy"/"dx" = -sqrt((1 - y^2)/(1 - x^2)`.
If x sin (a + y) + sin a . cos (a + y) = 0, then show that `"dy"/"dx" = (sin^2(a + y))/(sina)`.
If `x = e^(x/y)`, then show that `"dy"/"dx" = (x - y)/(xlogx)`
Differentiate log `[(sqrt(1 + x^2) + x)/(sqrt(1 + x^2 - x)]]` w.r.t. cos (log x).
If y2 = a2cos2x + b2sin2x, show that `y + (d^2y)/(dx^2) = (a^2b^2)/y^3`
Find `"dy"/"dx"` if, xy = log (xy)
Solve the following:
If `"x"^5 * "y"^7 = ("x + y")^12` then show that, `"dy"/"dx" = "y"/"x"`
If y = `("x" + sqrt("x"^2 - 1))^"m"`, then `("x"^2 - 1) "dy"/"dx"` = ______.
If `"x"^"a"*"y"^"b" = ("x + y")^("a + b")`, then show that `"dy"/"dx" = "y"/"x"`
If `sqrt(x) + sqrt(y) = sqrt("a")`, then `("d"y)/("d"x)` is ______
`(dy)/(dx)` of `2x + 3y = sin x` is:-
Differentiate w.r.t x (over no. 24 and 25) `e^x/sin x`
If log(x + y) = log(xy) + a then show that, `dy/dx = (-y^2)/x^2`
Find `dy/dx` if, x = `e^(3t)`, y = `e^sqrtt`
Solve the following.
If log(x + y) = log(xy) + a then show that, `dy/dx = (-y^2)/x^2`
Find `dy/dx"if", x= e^(3t), y=e^sqrtt`
