Commerce (English Medium) कक्षा १२ - CBSE Question Bank Solutions for Mathematics

If x^y = e^x–y, prove that `("d"y)/("d"x) = logx/(1 + logx)^2`

The derivative of log₁₀x w.r.t. x is ______.

If x = `e^(x/y)`, then prove that `dy/dx = (x - y)/(xlogx)`.

If y^x = e^{y – x}, prove that `"dy"/"dx" = (1 + log y)^2/logy`

If y = `(cos x)^((cos x)^((cosx)....oo)`, show that `"dy"/"dx" = (y^2 tanx)/(y log cos x - 1)`

Find `"dy"/"dx"`, if y = `x^tanx + sqrt((x^2 + 1)/2)`

Find `int_0^1 x(tan^-1x)  "d"x`

Evaluate the following:

`int (sin^-1 x)/((1 - x)^(3/2)) "d"x`

Evaluate the following:

`int ((cos 5x + cos 4x))/(1 - 2 cos 3x) "d"x`

Evaluate the following:

`int_0^1 x log(1 + 2x)  "d"x`

Evaluate the following:

`int_0^pi x log sin x "d"x`

`int "dx"/(sin(x - "a")sin(x - "b"))` is equal to ______.

`int tan^-1 sqrt(x)  "d"x` is equal to ______.

The vector having initial and terminal points as (2, 5, 0) and (–3, 7, 4), respectively is ______.

If `"y" = ("x" + sqrt(1 + "x"^2))^"n",  "then" (1 + "x"^2)  ("d"^2 "y")/"dx"^2 + "x" ("dy")/("dx")` is ____________.

If `"y = a"^"x", "b"^(2"x" -1), "then" ("d"^2"y")/"dx"^2` is ____________.

If `"y" = (varphi "n x")/"x",` then the value of y'' (e) is ____________.

If `"x" = "a" ("cos"  theta + theta  "sin"  theta), "y = a" ("sin"  theta - theta  "cos"  theta), "then" ("d"^2 "y")/("dx"^2) =` ____________.

If `"y"^2 = "ax"^2 + "bx + c", "then"  "d"/"dx" ("y"^3 "y"_"z") =` ____________.

If `sqrt(("x + y")) + sqrt (("y - x")) = "a", "then"  "dy"/"dx" =` ____________.