PUC Science 2nd PUC Class 12 - Karnataka Board PUC Question Bank Solutions for Mathematics

If \[y = \log \sqrt{\tan x}\] then the value of \[\frac{dy}{dx}\text { at }x = \frac{\pi}{4}\] is given by __________ .

If \[\sin^{- 1} \left( \frac{x^2 - y^2}{x^2 + y^2} \right) = \text { log a then } \frac{dy}{dx}\] is equal to _____________ .

If \[\sin y = x \cos \left( a + y \right), \text { then } \frac{dy}{dx}\] is equal to ______________ .

If \[y = \log \left( \frac{1 - x^2}{1 + x^2} \right), \text { then } \frac{dy}{dx} =\] __________ .

If \[y = \sqrt{\sin x + y}, \text { then }\frac{dy}{dx} \text { equals }\] ______________ .

If \[y = \tan^{- 1} \left( \frac{\sin x + \cos x}{\cos x - \sin x} \right), \text { then  } \frac{dy}{dx}\] is equal to ___________ .

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 \[\frac{d^2 y}{d x^2} + y = 0\] ?