Prove that the function \(f(x)=5x-3\) is continuous at \(x=0\), at \(x=-3\) and at \(x=5\).
Examine the continuity of the function \(f(x)=2x^2-1\) at \(x=3\).
\(f\) is continuous at \(x=3\).
Examine the following functions for continuity:
(a) \(f(x)=x-5\)
(b) \(f(x)=\frac{1}{x-5},\; x\ne 5\)
(c) \(f(x)=\frac{x^2-25}{x+5},\; x\ne -5\)
(d) \(f(x)=|x-5|\)
(a), (b), (c) and (d) are all continuous functions.
Prove that the function \(f(x)=x^n\) is continuous at \(x=n\), where \(n\) is a positive integer.
Is the function \(f\) defined by
\[ f(x)=\begin{cases} x, & x\le 1 \\ 5, & x>1 \end{cases} \]
continuous at \(x=0\)? At \(x=1\)? At \(x=2\)?
\(f\) is continuous at \(x=0\) and \(x=2\); not continuous at \(x=1\).
Find all points of discontinuity of \(f\), where \(f\) is defined by
\[ f(x)=\begin{cases} 2x+3, & x\le 2 \\ 2x-3, & x>2 \end{cases} \]
Discontinuous at \(x=2\).
Find all points of discontinuity of \(f\), where \(f\) is defined by
\[ f(x)=\begin{cases} |x|+3, & x\le -3 \\ -2x, & -3
Discontinuous at \(x=3\).
Find all points of discontinuity of \(f\), where \(f\) is defined by
\[ f(x)=\begin{cases} \frac{|x|}{x}, & x\ne 0 \\ 0, & x=0 \end{cases} \]
Discontinuous at \(x=0\).
Find all points of discontinuity of \(f\), where \(f\) is defined by
\[ f(x)=\begin{cases} \frac{x}{|x|}, & x<0 \\ -1, & x\ge 0 \end{cases} \]
No point of discontinuity.
Find all points of discontinuity of \(f\), where \(f\) is defined by
\[ f(x)=\begin{cases} x+1, & x\ge 1 \\ x^2+1, & x<1 \end{cases} \]
No point of discontinuity.
Find all points of discontinuity of \(f\), where \(f\) is defined by
\[ f(x)=\begin{cases} x^3-3, & x\le 2 \\ x^2+1, & x>2 \end{cases} \]
No point of discontinuity.
Find all points of discontinuity of \(f\), where \(f\) is defined by
\[ f(x)=\begin{cases} x^{10}-1, & x\le 1 \\ x^2, & x>1 \end{cases} \]
\(f\) is discontinuous at \(x=1\).
Is the function defined by
\[ f(x)=\begin{cases} x+5, & x\le 1 \\ x-5, & x>1 \end{cases} \]
a continuous function?
\(f\) is not continuous at \(x=1\).
Discuss the continuity of the function \(f\), where \(f\) is defined by
\[ f(x)=\begin{cases} 3, & 0\le x\le 1 \\ 4, & 1
\(f\) is not continuous at \(x=1\) and \(x=3\).
Discuss the continuity of the function \(f\), where \(f\) is defined by
\[ f(x)=\begin{cases} 2x, & x<0 \\ 0, & 0\le x\le 1 \\ 4x, & x>1 \end{cases} \]
\(x=1\) is the only point of discontinuity.
Discuss the continuity of the function \(f\), where \(f\) is defined by
\[ f(x)=\begin{cases} -2, & x\le -1 \\ 2x, & -1
Continuous
Find the relationship between \(a\) and \(b\) so that the function \(f\) defined by
\[ f(x)=\begin{cases} ax+1, & x\le 3 \\ bx+3, & x>3 \end{cases} \]
is continuous at \(x=3\).
\(a=b+\frac{2}{3}\)
For what value of \(\lambda\) is the function defined by
\[ f(x)=\begin{cases} \lambda(x^2-2x), & x\le 0 \\ 4x+1, & x>0 \end{cases} \]
continuous at \(x=0\)? What about continuity at \(x=1\)?
For no value of \(\lambda\), \(f\) is continuous at \(x=0\) but \(f\) is continuous at \(x=1\) for any value of \(\lambda\).
Show that the function defined by \(g(x)=x-[x]\) is discontinuous at all integral points. Here \([x]\) denotes the greatest integer less than or equal to \(x\).
Is the function defined by \(f(x)=x^2-\sin x+5\) continuous at \(x=\pi\)?
\(f\) is continuous at \(x=\pi\).
Discuss the continuity of the following functions:
(a) \(f(x)=\sin x+\cos x\)
(b) \(f(x)=\sin x-\cos x\)
(c) \(f(x)=\sin x\cdot\cos x\)
(a), (b) and (c) are all continuous
Discuss the continuity of the cosine, cosecant, secant and cotangent functions.
Cosine function is continuous for all \(x\in\mathbb{R}\); cosecant is continuous except for \(x=n\pi,\; n\in\mathbb{Z}\); secant is continuous except for \(x=\frac{(2n+1)\pi}{2},\; n\in\mathbb{Z}\) and cotangent function is continuous except for \(x=n\pi,\; n\in\mathbb{Z}\).
Find all points of discontinuity of \(f\), where
\[ f(x)=\begin{cases} \frac{\sin x}{x}, & x<0 \\ x+1, & x\ge 0 \end{cases} \]
There is no point of discontinuity.
Determine if \(f\) defined by
\[ f(x)=\begin{cases} x^2\sin\frac{1}{x}, & x\ne 0 \\ 0, & x=0 \end{cases} \]
is a continuous function.
Examine the continuity of \(f\), where \(f\) is defined by
\[ f(x)=\begin{cases} \sin x-\cos x, & x\ne 0 \\ -1, & x=0 \end{cases} \]
\(f\) is continuous for all \(x\in\mathbb{R}\).
Find the value of \(k\) so that the function \(f\) is continuous at \(x=\frac{\pi}{2}\), where
\[ f(x)=\begin{cases} \dfrac{k\cos x}{\pi-2x}, & x\ne \dfrac{\pi}{2} \\ 3, & x=\dfrac{\pi}{2} \end{cases} \]
\(k=6\)
Find the value of \(k\) so that the function \(f\) is continuous at \(x=2\), where
\[ f(x)=\begin{cases} kx^2, & x\le 2 \\ 3, & x>2 \end{cases} \]
\(k=\frac{3}{4}\)
Find the value of \(k\) so that the function \(f\) is continuous at \(x=\pi\), where
\[ f(x)=\begin{cases} kx+1, & x\le \pi \\ \cos x, & x>\pi \end{cases} \]
\(k=-\frac{2}{\pi}\)
Find the value of \(k\) so that the function \(f\) is continuous at \(x=5\), where
\[ f(x)=\begin{cases} kx+1, & x\le 5 \\ 3x-5, & x>5 \end{cases} \]
\(k=\frac{9}{5}\)
Find the values of \(a\) and \(b\) such that the function defined by
\[ f(x)=\begin{cases} 5, & x\le 2 \\ ax+b, & 2 is a continuous function.
\(a=2,\; b=1\)
Show that the function defined by \(f(x)=\cos(x^2)\) is a continuous function.
Show that the function defined by \(f(x)=|\cos x|\) is a continuous function.
Examine that \(\sin|x|\) is a continuous function.
Find all the points of discontinuity of \(f\) defined by \(f(x)=|x|-|x+1|\).
There is no point of discontinuity.
Differentiate with respect to \(x\): \(\sin(x^2+5)\).
\(2x\cos(x^2+5)\)
Differentiate with respect to \(x\): \(\cos(\sin x)\).
\(-\cos x\,\sin(\sin x)\)
Differentiate with respect to \(x\): \(\sin(ax+b)\).
\(a\cos(ax+b)\)
Differentiate with respect to \(x\): \(\sec(\tan(\sqrt{x}))\).
\(\dfrac{\sec(\tan\sqrt{x})\,\tan(\tan\sqrt{x})\,\sec^2\sqrt{x}}{2\sqrt{x}}\)
Differentiate with respect to \(x\): \(\dfrac{\sin(ax+b)}{\cos(cx+d)}\).
\(a\cos(ax+b)\sec(cx+d)+c\sin(ax+b)\tan(cx+d)\sec(cx+d)\)
Differentiate with respect to \(x\): \(\cos(x^3)\cdot \sin^2(x^5)\).
\(10x^4\sin(x^5)\cos(x^5)\cos(x^3)-3x^2\sin(x^3)\sin^2(x^5)\)
Differentiate with respect to \(x\): \(2\sqrt{\cot(x^2)}\).
\(\dfrac{-2\sqrt{2}\,x}{\sin(x^2)\sqrt{\sin(2x^2)}}\)
Differentiate with respect to \(x\): \(\cos(\sqrt{x})\).
\(\dfrac{-\sin\sqrt{x}}{2\sqrt{x}}\)
Prove that the function \(f\) given by \(f(x)=|x-1|\), \(x\in\mathbb{R}\), is not differentiable at \(x=1\).
Prove that the greatest integer function defined by \(f(x)=[x]\), \(0<x<3\), is not differentiable at \(x=1\) and \(x=2\).
Find \(\dfrac{dy}{dx}\) if \(2x+3y=\sin x\).
\(\dfrac{dy}{dx}=\dfrac{\cos x-2}{3}\)
Find \(\dfrac{dy}{dx}\) if \(2x+3y=\sin y\).
\(\dfrac{dy}{dx}=\dfrac{2}{\cos y-3}\)
Find \(\dfrac{dy}{dx}\) if \(ax+by^2=\cos y\).
\(\dfrac{dy}{dx}=-\dfrac{a}{2by+\sin y}\)
Find \(\dfrac{dy}{dx}\) if \(xy+y^2=\tan x+y\).
\(\dfrac{dy}{dx}=\dfrac{\sec^2 x-y}{x+2y-1}\)
Find \(\dfrac{dy}{dx}\) if \(x^2+xy+y^2=100\).
\(\dfrac{dy}{dx}=-\dfrac{2x+y}{x+2y}\)
Find \(\dfrac{dy}{dx}\) if \(x^3+x^2y+xy^2+y^3=81\).
\(\dfrac{dy}{dx}=-\dfrac{3x^2+2xy+y^2}{x^2+2xy+3y^2}\)
Find \(\dfrac{dy}{dx}\) if \(\sin^2 y+\cos(xy)=k\).
\(\dfrac{dy}{dx}=\dfrac{y\sin(xy)}{\sin(2y)-x\sin(xy)}\)
Find \(\dfrac{dy}{dx}\) if \(\sin^2 x+\cos^2 y=1\).
\(\dfrac{dy}{dx}=\dfrac{\sin 2x}{\sin 2y}\)
Find \(\dfrac{dy}{dx}\) if \(y=\sin^{-1}\!\left(\dfrac{2x}{1+x^2}\right)\).
\(\dfrac{dy}{dx}=\dfrac{2}{1+x^2}\)
Find \(\dfrac{dy}{dx}\) if \(y=\tan^{-1}\!\left(\dfrac{3x-x^3}{1-3x^2}\right)\), for \(-\dfrac{1}{\sqrt{3}}<x<\dfrac{1}{\sqrt{3}}\).
\(\dfrac{dy}{dx}=\dfrac{3}{1+x^2}\)
Find \(\dfrac{dy}{dx}\) if \(y=\cos^{-1}\!\left(\dfrac{1-x^2}{1+x^2}\right)\), for \(0<x<1\).
\(\dfrac{dy}{dx}=\dfrac{2}{1+x^2}\)
Find \(\dfrac{dy}{dx}\) if \(y=\sin^{-1}\!\left(\dfrac{1-x^2}{1+x^2}\right)\), for \(0<x<1\).
\(\dfrac{dy}{dx}=-\dfrac{2}{1+x^2}\)
Find \(\dfrac{dy}{dx}\) if \(y=\cos^{-1}\!\left(\dfrac{2x}{1+x^2}\right)\), for \(-1<x<1\).
\(\dfrac{dy}{dx}=-\dfrac{2}{1+x^2}\)
Find \(\dfrac{dy}{dx}\) if \(y=\sin^{-1}\!\left(2x\sqrt{1-x^2}\right)\), for \(-\dfrac{1}{\sqrt{2}}<x<\dfrac{1}{\sqrt{2}}\).
\(\dfrac{dy}{dx}=\dfrac{2}{\sqrt{1-x^2}}\)
Find \(\dfrac{dy}{dx}\) if \(y=\sec^{-1}\!\left(\dfrac{1}{2x^2-1}\right)\), for \(0<x<\dfrac{1}{\sqrt{2}}\).
\(\dfrac{dy}{dx}=-\dfrac{2}{\sqrt{1-x^2}}\)
Differentiate \(\dfrac{e^x}{\sin x}\) with respect to \(x\).
\(\dfrac{e^x(\sin x-\cos x)}{\sin^2 x}\), \(x \ne n\pi,\; n\in \mathbb{Z}\).
Differentiate \(e^{\sin^{-1}x}\) with respect to \(x\).
\(\dfrac{e^{\sin^{-1}x}}{\sqrt{1-x^2}}\), \(x\in(-1,1)\).
Differentiate \(e^{x^3}\) with respect to \(x\).
\(3x^2e^{x^3}\).
Differentiate \(\sin\big(\tan^{-1}(e^{-x})\big)\) with respect to \(x\).
\(-\dfrac{e^{-x}\cos\big(\tan^{-1}(e^{-x})\big)}{1+e^{-2x}}\).
Differentiate \(\log(\cos e^x)\) with respect to \(x\).
\(-e^x\tan(e^x)\), \(e^x\ne(2n+1)\dfrac{\pi}{2},\; n\in\mathbb{N}\).
Differentiate \(e^x + e^{x^2} + \cdots + e^{x^5}\) with respect to \(x\).
\(e^x + 2xe^{x^2} + 3x^2e^{x^3} + 4x^3e^{x^4} + 5x^4e^{x^5}\).
Differentiate \(\sqrt{e^{\sqrt{x}}}\) with respect to \(x\), where \(x>0\).
\(\dfrac{e^{\sqrt{x}}}{4\sqrt{x}e^{\sqrt{x}}}\), \(x>0\).
Differentiate \(\log(\log x)\) with respect to \(x\), where \(x>1\).
\(\dfrac{1}{x\log x}\), \(x>1\).
Differentiate \(\dfrac{\cos x}{\log x}\) with respect to \(x\), where \(x>0\).
\(-\dfrac{x\sin x\cdot\log x+\cos x}{x(\log x)^2}\), \(x>0\).
Differentiate \(\cos(\log x + e^x)\) with respect to \(x\), where \(x>0\).
\(-\dfrac{1}{x}+e^x\sin(\log x+e^x)\), \(x>0\).
Differentiate: \(\cos x\cdot \cos 2x\cdot \cos 3x\)
\(-\cos x\,\cos 2x\,\cos 3x\,[\tan x+2\tan 2x+3\tan 3x]\)
Differentiate: \(\sqrt{\dfrac{(x-1)(x-2)}{(x-3)(x-4)(x-5)}}\)
\(\dfrac12\sqrt{\dfrac{(x-1)(x-2)}{(x-3)(x-4)(x-5)}}\left[\dfrac1{x-1}+\dfrac1{x-2}-\dfrac1{x-3}-\dfrac1{x-4}-\dfrac1{x-5}\right]\)
Differentiate: \((\log x)^{\cos x}\)
\((\log x)^{\cos x}\left[\dfrac{\cos x}{x\log x}-\sin x\,\log(\log x)\right]\)
Differentiate: \(x^x-2^{\sin x}\)
\(x^x(1+\log x)-2^{\sin x}\,\cos x\,\log 2\)
Differentiate: \((x+3)^2\cdot (x+4)^3\cdot (x+5)^4\)
\((x+3)(x+4)^2(x+5)^3(9x^2+70x+133)\)
Differentiate: \(\left(x+\dfrac1x\right)^x+x^{\left(1+\frac1x\right)}\)
\(\left(x+\dfrac1x\right)^x\left[\dfrac{x^2-1}{x^2+1}+\log\left(x+\dfrac1x\right)\right]+x^{\left(1+\frac1x\right)}\left(\dfrac{x+1-\log x}{x^2}\right)\)
Differentiate: \((\log x)^x+x^{\log x}\)
\((\log x)^{x-1}\left[1+\log x\,\log(\log x)\right]+2x^{\log x-1}\,\log x\)
Differentiate: \((\sin x)^x+\sin^{-1}\sqrt{x}\)
\((\sin x)^x(x\cot x+\log\sin x)+\dfrac1{2\sqrt{x-x^2}}\)
Differentiate: \(x^{\sin x}+(\sin x)^{\cos x}\)
\(x^{\sin x}\left[\dfrac{\sin x}{x}+\cos x\,\log x\right]+(\sin x)^{\cos x}\left[\cos x\cot x-\sin x\,\log(\sin x)\right]\)
Differentiate: \(x^{x\cos x}+\dfrac{x^2+1}{x^2-1}\)
\(x^{x\cos x}\left[\cos x(1+\log x)-x\sin x\,\log x\right]-\dfrac{4x}{(x^2-1)^2}\)
Differentiate: \((x\cos x)^x+(x\sin x)^{\frac1x}\)
\((x\cos x)^x\left[1-x\tan x+\log(x\cos x)\right]+(x\sin x)^{\frac1x}\left[\dfrac{x\cot x+1-\log(x\sin x)}{x^2}\right]\)
Find \(\dfrac{dy}{dx}\) if \(x^y+y^x=1\).
\(-\dfrac{y\,x^{y-1}+y^x\log y}{x^y\log x+x\,y^{x-1}}\)
Find \(\dfrac{dy}{dx}\) if \(y^x=x^y\).
\(\dfrac{y}{x}\left(\dfrac{y-x\log y}{x-y\log x}\right)\)
Find \(\dfrac{dy}{dx}\) if \((\cos x)^y=(\cos y)^x\).
\(\dfrac{y\tan x+\log\cos y}{x\tan y+\log\cos x}\)
Find \(\dfrac{dy}{dx}\) if \(xy=e^{(x-y)}\).
\(\dfrac{y(x-1)}{x(y+1)}\)
Find the derivative of the function \(f(x)=(1+x)(1+x^2)(1+x^4)(1+x^8)\) and hence find \(f'(1)\).
\((1+x)(1+x^2)(1+x^4)(1+x^8)\left[\dfrac1{1+x}+\dfrac{2x}{1+x^2}+\dfrac{4x^3}{1+x^4}+\dfrac{8x^7}{1+x^8}\right]\); \(f'(1)=120\)
Differentiate \((x^2-5x+8)(x^3+7x+9)\) in three ways: (i) by using product rule (ii) by expanding the product (iii) by logarithmic differentiation. Do they all give the same answer?
\(5x^4-20x^3+45x^2-52x+11\)
If \(u\), \(v\) and \(w\) are functions of \(x\), show that \(\dfrac{d}{dx}(u\cdot v\cdot w)=\dfrac{du}{dx}\,v\,w+u\,\dfrac{dv}{dx}\,w+u\,v\,\dfrac{dw}{dx}\).
\(\dfrac{d}{dx}(uvw)=\dfrac{du}{dx}vw+u\dfrac{dv}{dx}w+uv\dfrac{dw}{dx}\)
If \(x = 2at^2\) and \(y = at^4\), find \(\dfrac{dy}{dx}\) without eliminating the parameter.
\(\dfrac{dy}{dx} = t^2\).
If \(x = a\cos\theta\) and \(y = b\cos\theta\), find \(\dfrac{dy}{dx}\) without eliminating the parameter.
\(\dfrac{dy}{dx} = \dfrac{b}{a}\).
If \(x = \sin t\) and \(y = \cos 2t\), find \(\dfrac{dy}{dx}\) without eliminating the parameter.
\(\dfrac{dy}{dx} = -4\sin t\).
If \(x = 4t\) and \(y = \dfrac{4}{t}\), find \(\dfrac{dy}{dx}\) without eliminating the parameter.
\(\dfrac{dy}{dx} = -\dfrac{1}{t^2}\).
If \(x = \cos\theta - \cos 2\theta\) and \(y = \sin\theta - \sin 2\theta\), find \(\dfrac{dy}{dx}\) without eliminating the parameter.
\(\dfrac{dy}{dx} = \dfrac{\cos\theta - 2\cos 2\theta}{2\sin 2\theta - \sin\theta}\).
If \(x = a(\theta - \sin\theta)\) and \(y = a(1 + \cos\theta)\), find \(\dfrac{dy}{dx}\) without eliminating the parameter.
\(\dfrac{dy}{dx} = -\cot\left(\dfrac{\theta}{2}\right)\).
If \(x = \dfrac{\sin^3 t}{\sqrt{\cos 2t}}\) and \(y = \dfrac{\cos^3 t}{\sqrt{\cos 2t}}\), find \(\dfrac{dy}{dx}\) without eliminating the parameter.
\(\dfrac{dy}{dx} = -\cot 3t\).
If \(x = a\left(\cos t + \log\tan\dfrac{t}{2}\right)\) and \(y = a\sin t\), find \(\dfrac{dy}{dx}\) without eliminating the parameter.
\(\dfrac{dy}{dx} = \tan t\).
If \(x = a\sec\theta\) and \(y = b\tan\theta\), find \(\dfrac{dy}{dx}\) without eliminating the parameter.
\(\dfrac{dy}{dx} = \dfrac{b}{a}\cosec\theta\).
If \(x = a(\cos\theta + \theta\sin\theta)\) and \(y = a(\sin\theta - \theta\cos\theta)\), find \(\dfrac{dy}{dx}\) without eliminating the parameter.
\(\dfrac{dy}{dx} = \tan\theta\).
If \(x = \sqrt{a^{\sin^{-1} t}}\) and \(y = \sqrt{a^{\cos^{-1} t}}\), show that \(\dfrac{dy}{dx} = -\dfrac{y}{x}\).
\(\dfrac{dy}{dx} = -\dfrac{y}{x}\).
Find the second order derivative of the function \(y = x^2 + 3x + 2\).
\(\dfrac{d^2y}{dx^2} = 2\)
Find the second order derivative of the function \(y = x^{20}\).
\(\dfrac{d^2y}{dx^2} = 380x^{18}\)
Find the second order derivative of the function \(y = x\cdot \cos x\).
\(\dfrac{d^2y}{dx^2} = -x\cos x - 2\sin x\)
Find the second order derivative of the function \(y = \log x\).
\(\dfrac{d^2y}{dx^2} = -\dfrac{1}{x^2}\)
Find the second order derivative of the function \(y = x^3\log x\).
\(\dfrac{d^2y}{dx^2} = x(5 + 6\log x)\)
Find the second order derivative of the function \(y = e^x\sin 5x\).
\(\dfrac{d^2y}{dx^2} = 2e^x(5\cos 5x - 12\sin 5x)\)
Find the second order derivative of the function \(y = e^{6x}\cos 3x\).
\(\dfrac{d^2y}{dx^2} = 9e^{6x}(3\cos 3x - 4\sin 3x)\)
Find the second order derivative of the function \(y = \tan^{-1}x\).
\(\dfrac{d^2y}{dx^2} = -\dfrac{2x}{(1 + x^2)^2}\)
Find the second order derivative of the function \(y = \log(\log x)\).
\(\dfrac{d^2y}{dx^2} = -\dfrac{1 + \log x}{(x\log x)^2}\)
Find the second order derivative of the function \(y = \sin(\log x)\).
\(\dfrac{d^2y}{dx^2} = -\dfrac{\sin(\log x) + \cos(\log x)}{x^2}\)
If \(y = \cos^{-1}x\), find \(\dfrac{d^2y}{dx^2}\) in terms of \(y\) alone.
\(\dfrac{d^2y}{dx^2} = -\cot y\,\cosec^2 y\)
Differentiate w.r.t. \(x\) the function \((3x^2-9x+5)^9\).
\(27(3x^2-9x+5)^8(2x-3)\)
Differentiate w.r.t. \(x\) the function \(\sin^3 x+\cos^6 x\).
\(3\sin x\cos x\,(\sin x-2\cos^4 x)\)
Differentiate w.r.t. \(x\) the function \((5x)^{3\cos 2x}\).
\((5x)^{3\cos 2x}\left[\dfrac{3\cos 2x}{x}-6\sin 2x\,\log(5x)\right]\)
Differentiate w.r.t. \(x\) the function \(\sin^{-1}(x\sqrt{x})\), for \(0\le x\le 1\).
\(\dfrac{3}{2}\sqrt{\dfrac{x}{1-x^3}}\)
Differentiate w.r.t. \(x\) the function \(\dfrac{\cos^{-1}(x/2)}{\sqrt{2x+7}}\), for \(-2<x<2\).
\(-\left[\dfrac{1}{\sqrt{4-x^2}\,\sqrt{2x+7}}+\dfrac{\cos^{-1}(x/2)}{(2x+7)^{3/2}}\right]\)
Differentiate w.r.t. \(x\) the function \(\cot^{-1}\left[\dfrac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right]\), for \(0<x<\dfrac{\pi}{2}\).
\(\dfrac{1}{2}\)
Differentiate w.r.t. \(x\) the function \((\log x)^{\log x}\), for \(x>1\).
\((\log x)^{\log x}\left[\dfrac{1}{x}+\dfrac{\log(\log x)}{x}\right]\)
Differentiate w.r.t. \(x\) the function \(\cos(a\cos x+b\sin x)\), where \(a\) and \(b\) are constants.
\((a\sin x-b\cos x)\,\sin(a\cos x+b\sin x)\)
Differentiate w.r.t. \(x\) the function \((\sin x-\cos x)^{(\sin x-\cos x)}\), for \(\dfrac{\pi}{4}<x<\dfrac{3\pi}{4}\).
\((\sin x-\cos x)^{(\sin x-\cos x)}(\cos x+\sin x)\left(1+\log(\sin x-\cos x)\right)\)
Differentiate w.r.t. \(x\) the function \(x^x+x^a+a^x+a^a\), where \(a>0\) is fixed and \(x>0\).
\(x^x(1+\log x)+a\,x^{a-1}+a^x\log a\)
Differentiate w.r.t. \(x\) the function \(x^{x^2-3}+(x-3)^{x^2}\), for \(x>3\).
\(x^{x^2-3}\left[\frac{x^2-3}{x}+2x\log x\right]+(x-3)^{x^2}\left[\frac{x^2}{x-3}+2x\log(x-3)\right]\)
Find \(\frac{dy}{dx}\), if \(y=12(1-\cos t)\), \(x=10(t-\sin t)\), \(-\frac{\pi}{2}<t<\frac{\pi}{2}\).
\(\frac{6}{5}\cot\frac{t}{2}\)
Find \(\frac{dy}{dx}\), if \(y=\sin^{-1}x+\sin^{-1}\sqrt{1-x^2}\), \(0<x<1\).
\(0\)
If \(x\sqrt{1+y}+y\sqrt{1+x}=0\), for \(-1<x<1\), prove that \(\frac{dy}{dx}=-\frac{1}{(1+x)^2}\).
If \((x-a)^2+(y-b)^2=c^2\), for some \(c>0\), prove that
\(\dfrac{\left[1+\left(\frac{dy}{dx}\right)^2\right]^{3/2}}{\frac{d^2y}{dx^2}}\) is a constant independent of \(a\) and \(b\).
If \(\cos y=x\cos(a+y)\), with \(\cos a\neq \pm 1\), prove that \(\frac{dy}{dx}=\frac{\cos^2(a+y)}{\sin a}\).
If \(x=a(\cos t+t\sin t)\) and \(y=a(\sin t-t\cos t)\), find \(\frac{d^2y}{dx^2}\).
\(\frac{\sec^3 t}{a t}\), \(0<t<\frac{\pi}{2}\)
If \(f(x)=|x|^3\), show that \(f''(x)\) exists for all real \(x\) and find it.
Using the fact that \(\sin(A+B)=\sin A\cos B+\cos A\sin B\) and the differentiation, obtain the sum formula for cosines.
Does there exist a function which is continuous everywhere but not differentiable at exactly two points? Justify your answer.
If \(y=\begin{vmatrix} f(x) & g(x) & h(x) \\ l & m & n \\ a & b & c \end{vmatrix}\), prove that
\(\frac{dy}{dx}=\begin{vmatrix} f'(x) & g'(x) & h'(x) \\ l & m & n \\ a & b & c \end{vmatrix}\).
If \(y=e^{a\cos^{-1}x}\), \(-1\le x\le 1\), show that \((1-x^2)\frac{d^2y}{dx^2}-x\frac{dy}{dx}-a^2y=0\).