List of top Mathematics Questions on Continuity and differentiability asked in CBSE CLASS XII

Differentiate the functions with respect to x.
\(sec(tan(\sqrt x))\)

Differentiate the functions with respect to x.
\(cos(sin\ x)\)

Examine the applicability of Mean Value Theorem for all three functions given in the above exercise 2.
If \(x=\sqrt{a^{sin^{-1}t}},y=\sqrt{a^{cos^{-1}t}}\),show that \(\frac{dy}{dx}=\frac{-y}{x}\)
Differentiate \((x^5-5x+8)(x^3+7x+9)\) in three ways mentioned below
(i) By using product rule
(ii) By expanding the product to obtain a single polynomial.
(iii) By logarithmic differentiation.
Do they all give the same answer?

If y=eacos-1x,-1≤x≤1,show that (1-x2)\(\frac{d^2y}{dx^2}\)-x\(\frac{dy}{dx}\)-a2y=0

\(y=\begin{vmatrix}  f(x)&g(x)  &h(x) \\   l&m  &n \\   a&b  &c  \end{vmatrix}\),prove that \(\frac{dy}{dx}=\begin{vmatrix}  f'(x)&g'(x)  &h'(x) \\   l&m  &n \\   a&b  &c  \end{vmatrix}\)

Using the fact that sin (A+B)=sinAcosB+cosAsinB and the differentiation, obtain the sum formula for cosines.

Using mathematical induction prove that \(\frac{d}{dx}\)(xn)=nxn-1 for all positive integers n

If f(x)=|x|3, show that f"(x)exists for all real x, and find it.

If x=a(cost+tsint) and y=a(sint-tcost),find \(\frac{d^2y}{dx^2}\)

If cosy=xcos(a+y) with cosa≠±1,prove that \(\frac{dy}{dx}\)=\(\frac{cos^2(a+y)}{sin\,a}\)

\(x\sqrt{1+y}+y\sqrt{1+x}=0\), for -1<x<1,prove that \(\frac{dy}{dx}\)=\(-\)\(\frac{1}{(1+x)^2}\)

Find \(\frac{dy}{dx}\), if y=sin-1x+sin-1\(\sqrt{1-x^2}\), -1≤x≤1

Verify Mean Value Theorem,if \(f(x)=x^3-5x^2-3x\) in the interval \([a,b]\),where \(a=1\) and \(b=3\).Find all \(c∈(1,3)\) for which \(f'(c)=0\)
Verify Mean Value Theorem,if \(f(x)=x^2-4x-3\) in the interval \([a,b]\),where \(a=1\) and \(b=4\)
If \(f:[-5,5]→R\) is a differentiable function and if \(f'(x)\)does not vanish anywhere, then prove that \(f(-5)≠f(5)\)
Verify Rolle’s Theorem for the function \(f(x)=x^2+2x-8,x∈[-4,2]\)
If \(y=(tan^{-1}x)^2\),show that \((x^2+1)^2y_2+2x(x^2+1)y_1=2\)
If \(y=500e^{7x}+600e^{-7x}\),show that \(\frac{d^2y}{dx^2}=49y\)
If \(y=Ae^{mx}+Be^{nx}\),show that \(\frac{d^2y}{dx^2}-(m+n)\frac{dy}{dx}+mny=0\)
If \(y=3cos(logx)+4sin(logx)\),show that \(x^2y_2+xy_1+y=0\)
If \(y=cos^{-1}x\),find \(\frac{d^2y}{dx^2}\) in terms of \(y\) alone.
If \(y=5cosx-3sinx\),prove that \(\frac{d^2y}{dx^2}+y=0\)
Find the second order derivatives of the function
\(sin(logx)\)