RD Sharma solutions for Mathematics [English] Class 12 chapter 12 - Higher Order Derivatives [Latest edition]

Find the second order derivatives of the following function  x³ + tan x ?

Find the second order derivatives of the following function sin (log x) ?

Find the second order derivatives of the following function  log (sin x) ?

Find the second order derivatives of the following function e^x sin 5x  ?

Find the second order derivatives of the following function e^6x cos 3x  ?

Find the second order derivatives of the following function x³ log x ?

Find the second order derivatives of the following function tan⁻¹ x ?

Find the second order derivatives of the following function x cos x ?

Find the second order derivatives of the following function  log (log x)  ?

If y = e^−x cos x, show that \[\frac{d^2 y}{d x^2} = 2 e^{- x} \sin x\] ?

If y = x + tan x, show that  \[\cos^2 x\frac{d^2 y}{d x^2} - 2y + 2x = 0\] ?

If y = x³ log x, prove that \[\frac{d^4 y}{d x^4} = \frac{6}{x}\] ?

If y = log (sin x), prove that \[\frac{d^3 y}{d x^3} = 2 \cos \ x \ {cosec}^3 x\] ?

If y = 2 sin x + 3 cos x, show that \[\frac{d^2 y}{d x^2} + y = 0\] ?

If \[y = \frac{\log x}{x}\] show that \[\frac{d^2 y}{d x^2} = \frac{2 \log x - 3}{x^3}\] ?

If x = a sec θ, y = b tan θ, prove that \[\frac{d^2 y}{d x^2} = - \frac{b^4}{a^2 y^3}\] ?

If `x=a (cos t +t sint )and y= a(sint-cos t )`  Prove that `Sec^3 t/(at),0<t< pi/2` 

If y = e^x cos x, prove that \[\frac{d^2 y}{d x^2} = 2 e^x \cos \left( x + \frac{\pi}{2} \right)\] ?

If x = a cos θ, y = b sin θ, show that \[\frac{d^2 y}{d x^2} = - \frac{b^4}{a^2 y^3}\] ?

If x = a (1 − cos³θ), y = a sin³θ, prove that \[\frac{d^2 y}{d x^2} = \frac{32}{27a} \text { at } \theta = \frac{\pi}{6}\]?

If x = a (θ + sin θ), y = a (1 + cos θ), prove that \[\frac{d^2 y}{d x^2} = - \frac{a}{y^2}\] ?

If x = a (θ − sin θ), y = a (1 + cos θ) prove that, find \[\frac{d^2 y}{d x^2}\] ?

If x = a(1 − cos θ), y = a(θ + sin θ), prove that \[\frac{d^2 y}{d x^2} = - \frac{1}{a}\text { at } \theta = \frac{\pi}{2}\] ?

If x = a (1 + cos θ), y = a(θ + sin θ), prove that \[\frac{d^2 y}{d x^2} = \frac{- 1}{a}at \theta = \frac{\pi}{2}\]

If x = cos θ, y = sin³ θ, prove that \[y\frac{d^2 y}{d x^2} + \left( \frac{dy}{dx} \right)^2 = 3 \sin^2 \theta\left( 5 \cos^2 \theta - 1 \right)\] ?

If y = sin (sin x), prove that \[\frac{d^2 y}{d x^2} + \tan x \cdot \frac{dy}{dx} + y \cos^2 x = 0\] ?

If x = sin t, y = sin pt, prove that \[\left( 1 - x^2 \right)\frac{d^2 y}{d x^2} - x\frac{dy}{dx} + p^2 y = 0\] ?

If y = (sin⁻¹ x)², prove that (1 − x²)

\[\frac{d^2 y}{d x^2} - x\frac{dy}{dx} + p^2 y = 0\] ?

If \[y = e^{\tan^{- 1} x}\] prove that (1 + x²)y₂ + (2x − 1)y₁ = 0 ?

If y = 3 cos (log x) + 4 sin (log x), prove that x²y₂ + xy₁ + y = 0 ?

If \[y = e^{2x} \left( ax + b \right)\]  show that  \[y_2 - 4 y_1 + 4y = 0\] ?

If `x = sin(1/2 log y)` show that (1 − x²)y₂ − xy₁ − a²y = 0.

If y = tan⁻¹ x, show that \[\left( 1 + x^2 \right) \frac{d^2 y}{d x^2} + 2x\frac{dy}{dx} = 0\] ?

If \[y = \left[ \log \left( x + \sqrt{x^2 + 1} \right) \right]^2\] show that \[\left( 1 + x^2 \right)\frac{d^2 y}{d x^2} + x\frac{dy}{dx} = 2\] ?

If y = cot x show that \[\frac{d^2 y}{d x^2} + 2y\frac{dy}{dx} = 0\] ?

Find \[\frac{d^2 y}{d x^2}\] where \[y = \log \left( \frac{x^2}{e^2} \right)\] ?

If y = ae^2x + be^−x, show that, \[\frac{d^2 y}{d x^2} - \frac{dy}{dx} - 2y = 0\] ?

If y = e^x (sin x + cos x) prove that \[\frac{d^2 y}{d x^2} - 2\frac{dy}{dx} + 2y = 0\] ?

If y = cos⁻¹ x, find \[\frac{d^2 y}{d x^2}\] in terms of y alone ?

If  \[y = e^{a \cos^{- 1}} x\] ,prove that \[\left( 1 - x^2 \right)\frac{d^2 y}{d x^2} - x\frac{dy}{dx} - a^2 y = 0\] ?

If y = 500 e^7x + 600 e^−7x, show that \[\frac{d^2 y}{d x^2} = 49y\] ?

If x = 2 cos t − cos 2t, y = 2 sin t − sin 2t, find \[\frac{d^2 y}{d x^2}\text{ at } t = \frac{\pi}{2}\] ?

If x = 4z² + 5, y = 6z² + 7z + 3, find \[\frac{d^2 y}{d x^2}\] ?

If y log (1 + cos x), prove that \[\frac{d^3 y}{d x^3} + \frac{d^2 y}{d x^2} \cdot \frac{dy}{dx} = 0\] ?

If y = sin (log x), prove that \[x^2 \frac{d^2 y}{d x^2} + x\frac{dy}{dx} + y = 0\] ?

If y = 3 e^2x + 2 e^3x, prove that  \[\frac{d^2 y}{d x^2} - 5\frac{dy}{dx} + 6y = 0\] ?

If y = (cot⁻¹ x)², prove that y₂(x² + 1)² + 2x (x² + 1) y₁ = 2 ?

If y = cosec⁻¹ x, x >1, then show that \[x\left( x^2 - 1 \right)\frac{d^2 y}{d x^2} + \left( 2 x^2 - 1 \right)\frac{dy}{dx} = 0\] ?

\[\text { If x } = \cos t + \log \tan\frac{t}{2}, y = \sin t, \text { then find the value of } \frac{d^2 y}{d t^2} \text { and } \frac{d^2 y}{d x^2} \text { at } t = \frac{\pi}{4} \] ?

\[ \text { If x } = a \sin t \text { and y } = a\left( \cos t + \log \tan\frac{t}{2} \right), \text { find } \frac{d^2 y}{d x^2} \] ?

\[\text { If x } = a\left( \cos t + t \sin t \right) \text { and y} = a\left( \sin t - t \cos t \right),\text { then find the value of } \frac{d^2 y}{d x^2} \text { at } t = \frac{\pi}{4} \] ?

\[\text{ If x } = a\left( \cos t + \log \tan\frac{t}{2} \right) \text { and y } = a\left( \sin t \right), \text { evaluate } \frac{d^2 y}{d x^2} \text { at t } = \frac{\pi}{3} \] ?

\[\text { If x } = a\left( \cos2t + 2t \sin2t \right)\text {  and y } = a\left( \sin2t - 2t \cos2t \right), \text { then find } \frac{d^2 y}{d x^2} \] ?

If \[x = 3 \cos t - 2 \cos^3 t, y = 3\sin t - 2 \sin^3 t,\] find \[\frac{d^2 y}{d x^2} \] ?

\[\text { If x } = a \sin t - b \cos t, y = a \cos t + b \sin t, \text { prove that } \frac{d^2 y}{d x^2} = - \frac{x^2 + y^2}{y^3} \] ?

\[\text { Find A and B so that y = A } \sin3x + B \cos3x \text { satisfies the equation }\]

\[\frac{d^2 y}{d x^2} + 4\frac{d y}{d x} + 3y = 10 \cos3x \] ?

\[\text { If }y = A e^{- kt} \cos\left( pt + c \right), \text { prove that } \frac{d^2 y}{d t^2} + 2k\frac{d y}{d t} + n^2 y = 0, \text { where } n^2 = p^2 + k^2 \] ?

\[\text { If y } = x^n \left\{ a \cos\left( \log x \right) + b \sin\left( \log x \right) \right\}, \text { prove that } x^2 \frac{d^2 y}{d x^2} + \left( 1 - 2n \right)x\frac{d y}{d x} + \left( 1 + n^2 \right)y = 0 \] Disclaimer: There is a misprint in the question. It must be 

\[x^2 \frac{d^2 y}{d x^2} + \left( 1 - 2n \right)x\frac{d y}{d x} + \left( 1 + n^2 \right)y = 0\] instead of 1

\[x^2 \frac{d^2 y}{d x^2} + \left( 1 - 2n \right)\frac{d y}{d x} + \left( 1 + n^2 \right)y = 0\] ?

\[\text { If y } = a \left\{ x + \sqrt{x^2 + 1} \right\}^n + b \left\{ x - \sqrt{x^2 + 1} \right\}^{- n} , \text { prove that }\left( x^2 + 1 \right)\frac{d^2 y}{d x^2} + x\frac{d y}{d x} - n^2 y = 0 \]

Disclaimer: There is a misprint in the question,

\[\left( x^2 + 1 \right)\frac{d^2 y}{d x^2} + x\frac{d y}{d x} - n^2 y = 0\] must be written instead of

\[\left( x^2 - 1 \right)\frac{d^2 y}{d x^2} + x\frac{d y}{d x} - n^2 y = 0 \] ?

If y = a x^{n + 1} + bx⁻ⁿ and \[x^2 \frac{d^2 y}{d x^2} = \lambda y\]  then write the value of λ ?

If x = a cos nt − b sin nt and \[\frac{d^2 x}{dt} = \lambda x\]  then find the value of λ ?

If x = t² and y = t³, find \[\frac{d^2 y}{d x^2}\] ?

If x = 2at, y = at², where a is a constant, then find \[\frac{d^2 y}{d x^2} \text { at }x = \frac{1}{2}\] ?

If x = f(t) and y = g(t), then write the value of \[\frac{d^2 y}{d x^2}\] ?

If \[y = 1 - x + \frac{x^2}{2!} - \frac{x^3}{3!} + \frac{x^4}{4!}\] .....to ∞, then write  \[\frac{d^2 y}{d x^2}\] in terms of y ?

If y = x + e^x, find \[\frac{d^2 x}{d y^2}\] ?

If y = |x − x²|, then find \[\frac{d^2 y}{d x^2}\] ?

If \[y = \left| \log_e x \right|\] find\[\frac{d^2 y}{d x^2}\] ?

If x = a cos nt − b sin nt, then \[\frac{d^2 x}{d t^2}\] is 

If x = at², y = 2 at, then \[\frac{d^2 y}{d x^2} =\] 

If y = axⁿ⁺¹ + bx⁻ⁿ, then \[x^2 \frac{d^2 y}{d x^2} =\] 

\[\frac{d^{20}}{d x^{20}} \left( 2 \cos x \cos 3 x \right) =\]

If x = t², y = t³, then \[\frac{d^2 y}{d x^2} =\] 

If y = a + bx², a, b arbitrary constants, then

If f(x) = (cos x + i sin x) (cos 2x + i sin 2x) (cos 3x + i sin 3x) ...... (cos nx + i sin nx) and f(1) = 1, then f'' (1) is equal to

If y = a sin mx + b cos mx, then \[\frac{d^2 y}{d x^2}\]   is equal to

If \[f\left( x \right) = \frac{\sin^{- 1} x}{\sqrt{1 - x^2}}\] then (1 − x)² f '' (x) − xf(x) =

If \[y = \tan^{- 1} \left\{ \frac{\log_e \left( e/ x^2 \right)}{\log_e \left( e x^2 \right)} \right\} + \tan^{- 1} \left( \frac{3 + 2 \log_e x}{1 - 6 \log_e x} \right)\], then \[\frac{d^2 y}{d x^2} =\]

Let f(x) be a polynomial. Then, the second order derivative of f(e^x) is

If y = a cos (log_e x) + b sin (log_e x), then x² y₂ + xy₁ =

If x = 2 at, y = at², where a is a constant, then \[\frac{d^2 y}{d x^2} \text { at x } = \frac{1}{2}\] is 

If x = f(t) and y = g(t), then \[\frac{d^2 y}{d x^2}\] is equal to

If y = sin (m sin⁻¹ x), then (1 − x²) y₂ − xy₁ is equal to

If y = (sin⁻¹ x)², then (1 − x²)y₂ is equal to

If y = e^tan x, then (cos² x)y₂ =

If \[y = \frac{ax + b}{x^2 + c}\] then (2xy₁ + y)y₃ = 

If \[y = \log_e \left( \frac{x}{a + bx} \right)^x\] then x³ y₂ =

If x = f(t) cos t − f' (t) sin t and y = f(t) sin t + f'(t) cos t, then\[\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2 =\]

If \[y^\frac{1}{n} + y^{- \frac{1}{n}} = 2x, \text { then find } \left( x^2 - 1 \right) y_2 + x y_1 =\] ?

If \[\frac{d}{dx}\left[ x^n - a_1 x^{n - 1} + a_2 x^{n - 2} + . . . + \left( - 1 \right)^n a_n \right] e^x = x^n e^x\] then the value of a_r, 0 < r ≤ n, is equal to 

If y = xⁿ⁻¹ log x then x² y₂ + (3 − 2n) xy₁ is equal to

If xy − log_e y = 1 satisfies the equation \[x\left( y y_2 + y_1^2 \right) - y_2 + \lambda y y_1 = 0\]

If y² = ax² + bx + c, then \[y^3 \frac{d^2 y}{d x^2}\] is