HSC Science (General) 12th Standard Board Exam - Maharashtra State Board Important Questions for Mathematics and Statistics

If y = f (x) is a differentiable function of x such that inverse function x = f ^–1(y) exists, then
prove that x is a differentiable function of y and 

`dx/dy=1/(dy/dx)`, Where `dy/dxne0`

Hence if `y=sin^-1x, -1<=x<=1 , -pi/2<=y<=pi/2`

then show that `dy/dx=1/sqrt(1-x^2)`, where  `|x|<1`

If `x = acos^3t`, `y = asin^3 t`,

Show that `(dy)/(dx) =- (y/x)^(1/3)`

Find `dy/dx` if `y = tan^(-1) ((5x+ 1)/(3-x-6x^2))`

If X = f(t) and Y = g(t) Are Differentiable Functions of t ,  then prove that y is a differentiable function of x and

`"dy"/"dx" =("dy"/"dt")/("dx"/"dt" ) , "where" "dx"/"dt" ≠ 0`

Hence find `"dy"/"dx"` if x = a cos² t and y = a sin² t.

If y = sin ^-1 `((8x)/(1 + 16x^2))`, find `(dy)/(dx)`

The total cost function of a firm is C = x² + 75x + 1600 for output x. Find the  output for which the average cost ls minimum. Is C_A= C_m at this output?  

Differentiate the following w.r.t.x:

tan[cos(sinx)]

Differentiate the following w.r.t. x: `x^(tan^(-1)x`

Differentiate the following w.r.t. x: x^e + x^x + e^x + e^e.

If `log_10((x^3 - y^3)/(x^3 + y^3))` = 2, show that `dy/dx = -(99x^2)/(101y^2)`.

If y = `log(x + sqrt(x^2 + a^2))^m`, show that `(x^2 + a^2)(d^2y)/(dx^2) + x "d"/"dx"` = 0.

If y is a function of x and log (x + y) = 2xy, then the value of y'(0) = ______.

If f(x) = log_x (log x) then f'(e) is ______

If y = `"e"^(1 + logx)` then find `("d"y)/("d"x)` 

If y = log [cos(x⁵)] then find `("d"y)/("d"x)`

Differentiate sin² (sin⁻¹(x²)) w.r. to x

Differentiate `cot^-1((cos x)/(1 + sinx))` w.r. to x

Differentiate `sin^-1((2cosx + 3sinx)/sqrt(13))` w.r. to x

If log₅ `((x^4 + "y"^4)/(x^4 - "y"^4))` = 2, show that `("dy")/("d"x) = (12x^3)/(13"y"^2)`

If y = `sqrt(cos x + sqrt(cos x + sqrt(cos x + ...... ∞)`, show that `("d"y)/("d"x) = (sin x)/(1 - 2y)`