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 \((1-x^2)\frac{d^2y}{dx^2}-x\frac{dy}{dx}-a^2y=0\).