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 \(\dfrac{d}{dx}(u\cdot v\cdot w)=\dfrac{du}{dx}\,v\,w+u\,\dfrac{dv}{dx}\,w+u\,v\,\dfrac{dw}{dx}\).
\(\dfrac{d}{dx}(uvw)=\dfrac{du}{dx}vw+u\dfrac{dv}{dx}w+uv\dfrac{dw}{dx}\)