HSC Commerce (English Medium) १२ वीं कक्षा - Maharashtra State Board Important Questions

If x = tan^-1t and y = t³ , find `(dy)/(dx)`.

Discuss extreme values of the function f(x) = x.logx

If e^x + e^y = e^{(x + y)}, then show that `dy/dx = -e^(y - x)`.

Find `dy/dx`if, y = `(x)^x + (a^x)`.

Find `"dy"/"dx"`, if x = e^3t, y = `"e"^((4"t" + 5))`

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

If x = t . log t, y = t^t, then show that `dy/dx - y = 0`.

If y = e^logx then `dy/dx` = ?

If y = log `("e"^"x"/"x"^2)`, then `"dy"/"dx" = ?` 

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 a^x is a^x log a.

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

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

Find `(dy)/(dx)`, if x^y = y^x 

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")`