Recent questions in Continuity and Differentiability

If $f (x) = |\; x\; |^3$ , show that $f ″(x)$ exists for all real $x$ and find it.

asked Dec 4, 2012 by thanvigandhi_1

1 answer

If $x = a (\cos t + t\sin t)$ and $y = a (\sin t - t \cos\: t)$ , find $\large\frac{d^2y}{dx^2}$

asked Dec 4, 2012 by thanvigandhi_1

1 answer

If $cos\: y = x \cos (a + y)$ , with $\cos$ $a \neq \pm 1$ , prove that $\large\frac{dy}{dx}=\frac{\cos^2(a + y)}{\sin \: a}$

asked Dec 4, 2012 by thanvigandhi_1

1 answer

If $( x-a)^2 + (y-b)^2 = c^2$ , for some $c > 0$ , prove that $\Large\frac{\begin{bmatrix} 1 + \left(\frac{dy}{dx}\right)^2 \\[0.3em] \end{bmatrix}^{\frac{\Large 3}{\Large 2}}}{\Large\frac{d^2y}{dx^2}}$ is a constant independent of $a$ and $b$ .

asked Nov 21, 2012 by thanvigandhi_1

1 answer

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

asked Nov 21, 2012 by thanvigandhi_1

1 answer

Differentiate w.r.t. x the function $cot^{-1} \begin{bmatrix} \frac{{\sqrt {1+sin \: x}} + {\sqrt{1-sin \: x}}} {{\sqrt {1+sin \: x}} - {\sqrt{1-sin \: x}}} \\[0.3em] \end{bmatrix}$ , where $0 < x < \frac{\pi}{2}$

asked Nov 21, 2012 by thanvigandhi_1

1 answer

Find $\large \frac{dy}{dx}$ , if $y = \sin^{-1}x + \sin^{-1} \: \sqrt {1-x^2} , -1 \leq x \leq 1$

asked Nov 21, 2012 by thanvigandhi_1

1 answer

Find $\large\frac{dy}{dx}$ , if $y = 12 ( 1 - \cos t ) , x = 10 ( t - \sin t ) , -\large\frac{\pi}{2}\normalsize < t <\large \frac{\pi}{2}$

asked Nov 21, 2012 by thanvigandhi_1

1 answer

Differentiate w.r.t. $x$ the function in $x^{\large x^2-3} + ( x - 3 )^{\large x^2}$ , for $x > 3$

asked Nov 21, 2012 by thanvigandhi_1

1 answer

Differentiate w.r.t. $x$ the function in $x^{\large x} + x^{\large a} + a^{\large x} + a^{\large a}$ , for some fixed $a > 0$ and $x > 0$

asked Nov 21, 2012 by thanvigandhi_1

1 answer

Differentiate w.r.t. $x$ the function in $(\sin x - \cos x )^{\large(\sin x - \cos x )} , \large\frac {\pi}{4}\normalsize < x < \large\frac{3\pi}{4}$

asked Nov 21, 2012 by thanvigandhi_1

1 answer

Differentiate w.r.t. $x$ the function in $\cos \: (a \cos x + b \sin x)$ , for some constant $a$ and $b$

asked Nov 21, 2012 by thanvigandhi_1

1 answer

Differentiate w.r.t. $x$ the function in $( \log x )^{\large\log x}, x > 1$

asked Nov 21, 2012 by thanvigandhi_1

1 answer

Differentiate w.r.t. $x$ the function in $\large\frac{\cos^{ -1}\Large\frac{ x}{ 2}}{\sqrt {2x + 7}}, \normalsize-2 < x < 2$

asked Nov 21, 2012 by thanvigandhi_1

1 answer

Differentiate w.r.t. $x$ the function in $sin^{-1}(x \: \sqrt x ), 0 \leq x \leq 1$

asked Nov 21, 2012 by thanvigandhi_1

1 answer

Differentiate w.r.t. $x$ the function in $(5x)^{\large 3 \cos 2x}$

asked Nov 21, 2012 by thanvigandhi_1

1 answer

Differentiate w.r.t. $x$ the function in $sin^3x + cos^6x$

asked Nov 21, 2012 by thanvigandhi_1

1 answer

Differentiate w.r.t. $x$ the function in $(3x^2 - 9x + 5)^9$

asked Nov 21, 2012 by thanvigandhi_1

1 answer

Examine the applicability of Mean Value Theorem for all three functions given in $(i)\;f (x) = [x] \;for\; x \: \in [5,9]$

asked Nov 21, 2012 by thanvigandhi_1

1 answer

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 $c \in (1,3)$ for which $f'(c) = 0$ .

asked Nov 21, 2012 by thanvigandhi_1

1 answer

Verify Mean Value Theorem, if $f (x) = x^2 - 4x - 3$ in the interval $[a,b]$ , where $a = 1$ and $b = 4$ .

asked Nov 21, 2012 by thanvigandhi_1

1 answer