Find the derivative of the following functions from first principle:
(i) \(-x\)
(ii) \((-x)^{-1}\)
(iii) \(\sin (x+1)\)
(iv) \(\cos \left(x-\tfrac{\pi}{8}\right)\)
(i) \(-1\)
(ii) \(\dfrac{1}{x^{2}}\)
(iii) \(\cos (x+1)\)
(iv) \(-\sin \left(x-\tfrac{\pi}{8}\right)\)
Find the derivative of \(x+a\).
1
Find the derivative of \((px+q)\left(\dfrac{r}{x}+s\right)\), where \(p,q,r,s\) are fixed non-zero constants.
\(-\dfrac{qr}{x^{2}}+ps\)
Find the derivative of \((ax+b)(cx+d)^{2}\).
\(2c(ax+b)(cx+d)+a(cx+d)^{2}\)
Find the derivative of \(\dfrac{ax+b}{cx+d}\).
\(\dfrac{ad-bc}{(cx+d)^{2}}\)
Find the derivative of \(\dfrac{1+\tfrac{1}{x}}{1-\tfrac{1}{x}}\).
\(-\dfrac{2}{(x-1)^{2}},\ x\neq 0,1\)
Find the derivative of \(\dfrac{1}{ax^{2}+bx+c}\).
\(-\dfrac{2ax+b}{(ax^{2}+bx+c)^{2}}\)
Find the derivative of \(\dfrac{ax+b}{px^{2}+qx+r}\).
\(-\dfrac{apx^{2}+2bpx+ar-bq}{(px^{2}+qx+r)^{2}}\)
Find the derivative of \(\dfrac{px^{2}+qx+r}{ax+b}\).
\(\dfrac{apx^{2}+2bpx+bq-ar}{(ax+b)^{2}}\)
Find the derivative of \(\dfrac{a}{x^{4}}-\dfrac{b}{x^{2}}+\cos x\).
\(-\dfrac{4a}{x^{5}}+\dfrac{2b}{x^{3}}-\sin x\)
Find the derivative of \(4\sqrt{x}-2\).
\(\dfrac{2}{\sqrt{x}}\)
Find the derivative of \((ax+b)^{n}\).
\(na(ax+b)^{n-1}\)
Find the derivative of \((ax+b)^{n}(cx+d)^{m}\).
\((ax+b)^{n-1}(cx+d)^{m-1}\big[mc(ax+b)+na(cx+d)\big]\)
Find the derivative of \(\sin(x+a)\).
\(\cos(x+a)\)
Find the derivative of \(\csc x\cot x\).
\(-\csc^{3}x-\csc x\cot^{2}x\)
Find the derivative of \(\dfrac{\cos x}{1+\sin x}\).
\(-\dfrac{1}{1+\sin x}\)
Find the derivative of \(\dfrac{\sin x+\cos x}{\sin x-\cos x}\).
\(-\dfrac{2}{(\sin x-\cos x)^{2}}\)
Find the derivative of \(\dfrac{\sec x-1}{\sec x+1}\).
\(\dfrac{2\sec x\tan x}{(\sec x+1)^{2}}\)
Find the derivative of \(\sin^{n}x\).
\(n\sin^{n-1}x\cos x\)
Find the derivative of \(\dfrac{a+b\sin x}{c+d\cos x}\).
\(\dfrac{bc\cos x+ad\sin x+bd}{(c+d\cos x)^{2}}\)
Find the derivative of \(\dfrac{\sin(x+a)}{\cos x}\).
\(\dfrac{\cos a}{\cos^{2}x}\)
Find the derivative of \(x^{4}(5\sin x-3\cos x)\).
\(x^{3}\big(5x\cos x+3x\sin x+20\sin x-12\cos x\big)\)
Find the derivative of \((x^{2}+1)\cos x\).
\(-x^{2}\sin x-\sin x+2x\cos x\)
Find the derivative of \((ax^{2}+\sin x)(p+q\cos x)\).
\(-q\sin x(ax^{2}+\sin x)+(p+q\cos x)(2ax+\cos x)\)
Find the derivative of \((x+\cos x)(x-\tan x)\).
\(-\tan^{2}x(x+\cos x)+(x-\tan x)(1-\sin x)\)
Find the derivative of \(\dfrac{4x+5\sin x}{3x+7\cos x}\).
\(\dfrac{35+15x\cos x+28\cos x+28x\sin x-15\sin x}{(3x+7\cos x)^{2}}\)
Find the derivative of \(\dfrac{x^{2}\cos\left(\tfrac{\pi}{4}\right)}{\sin x}\).
\(\dfrac{x\cos\left(\tfrac{\pi}{4}\right)(2\sin x-x\cos x)}{\sin^{2}x}\)
Find the derivative of \(\dfrac{x}{1+\tan x}\).
\(\dfrac{1+\tan x-x\sec^{2}x}{(1+\tan x)^{2}}\)
Find the derivative of \((x+\sec x)(x-\tan x)\).
\((x+\sec x)(1-\sec^{2}x)+(x-\tan x)(1+\sec x\tan x)\)
Find the derivative of \(\dfrac{x}{\sin^{n}x}\).
\(\dfrac{\sin x-nx\cos x}{\sin^{n+1}x}\)