HSC Commerce (English Medium) १२ वीं कक्षा - Maharashtra State Board Important Questions for Mathematics and Statistics

If x = 2at² , y = 4at, then `dy/dx = ?`

If x = `y + 1/y`, then `dy/dx` = ____.

If y = `"e"^"ax"`, then `"x" * "dy"/"dx" =`______.

State whether the following is True or False:

The derivative of `log_ax`, where a is constant is `1/(x.loga)`.

The derivative of f(x) = a^x, where a is constant is x.a^x-1.

The derivative of a^x is a^x log a.

Find the rate of change of demand (x) of a commodity with respect to its price (y) if y = `(5x + 7)/(2x - 13)`

If `x^7 * y^9 = (x + y)^16`, then show that `dy/dx = y/x`

If y = sec (tan⁻¹x), then `dy/dx` at x = 1 is ______.

Find `("d"y)/("d"x)`, if y = [log(log(logx))]² 

Find `(dy)/(dx)`, if x^y = y^x 

Find `("d"^2y)/("d"x^2)`, if y = `"e"^((2x + 1))`

Find `("d"y)/("d"x)`, if y = `root(3)(((3x - 1))/((2x + 3)(5 - x)^2)`

If x = `(4"t")/(1 + "t"^2)`, y = `3((1 - "t"^2)/(1 + "t"^2))`, then show that `("d"y)/("d"x) = (-9x)/(4y)` 

Find `("d"y)/("d"x)`, if x = e^m, y = `"e"^(sqrt("m"))`

Solution: Given, x = e^m and y = `"e"^(sqrt("m"))`

Now, y = `"e"^(sqrt("m"))`

Diff.w.r.to m,

`("d"y)/"dm" = "e"^(sqrt("m"))("d"square)/"dm"`

∴ `("d"y)/"dm" = "e"^(sqrt("m"))*1/(2sqrt("m"))`    .....(i)

Now, x = e^m

Diff.w.r.to m,

`("d"x)/"dm" = square`    .....(ii)

Now, `("d"y)/("d"x) = (("d"y)/("d"m))/square`

∴ `("d"y)/("d"x) = (("e"sqrt("m"))/square)/("e"^"m")`

∴  `("d"y)/("d"x) = ("e"^(sqrt("m")))/(2sqrt("m")*"e"^("m")`

If x = `sqrt(1 + u^2)`, y = `log(1 + u^2)`, then find `(dy)/(dx).`

If ax² + 2hxy + by² = 0, then prove that `(d^2y)/(dx^2)` = 0.

Solve the following differential equations:

x²ydx – (x³ – y³)dy = 0

Find `(d^2y)/(dy^2)`, if y = e^4x

`int 1/(4x^2 - 1) dx` = ______.