HSC Commerce (English Medium) 12th Standard Board Exam - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

If y = `e^(mtan^-1x)`, show that `(1 + x^2)(d^2y)/(dx^2) + (2x - m)"dy"/"dx"` = 0.

If x = cos t, y = e^mt, show that `(1 - x^2)(d^2y)/(dx^2) - x"dy"/"dx" - m^2y` = 0.

If y = x + tan x, show that `cos^2x.(d^2y)/(dx^2) - 2y + 2x` = 0.

If y = e^ax.sin(bx), show that y₂ – 2ay₁ + (a² + b²)y = 0.

If `sec^-1((7x^3 - 5y^3)/(7^3 + 5y^3)) = "m", "show"  (d^2y)/(dx^2)` = 0.

If 2y = `sqrt(x + 1) + sqrt(x - 1)`, show that 4(x² – 1)y₂ + 4xy₁ – y = 0.

If y = sin (m cos^–1x), then show that `(1 - x^2)(d^2y)/(dx^2) - x"dy"/"dx" + m^2y` = 0.

If x² + 6xy + y² = 10, show that `(d^2y)/(dx^2) = (80)/(3x + y)^3`.

If x = a sin t – b cos t, y = a cos t + b sin t, show that `(d^2y)/(dx^2) = -(x^2 + y^2)/(y^3)`.

Find the n^th derivative of the following : (ax + b)^m 

Find the n^th derivative of the following:

`(1)/x`

Find the n^th derivative of the following : e^ax+b 

Find the nth derivative of the following : a^px+q 

Find the n^th derivative of the following : cos x

Find the n^th derivative of the following : sin (ax + b)

Find the n^th derivative of the following : cos (3 – 2x)

Find the n^th derivative of the following : `(1)/(3x - 5)`

Find the n^th derivative of the following : y = e^ax . cos (bx + c)

Find the n^th derivative of the following:

y = e^8x . cos (6x + 7)

Choose the correct option from the given alternatives : 

Let `f(1) = 3, f'(1) = -(1)/(3), g(1) = -4 and g'(1) =-(8)/(3).` The derivative of `sqrt([f(x)]^2 + [g(x)]^2` w.r.t. x at x = 1 is