(1) Definition and notation : If \[y\] is a function of \[x\] and is differentiable with respect to \[x,\] then its derivative \[\frac{dy}{dx}\]can be found which is known as derivative of first order. If the first derivative \[\frac{dy}{dx}\] is also a differentiable function, then it can be further differentiated with respect to x and this derivative is denoted by \[{{d}^{2}}y/d{{x}^{2}}\], which is called the second derivative of \[y\] with respect to \[x\]. Further if \[\frac{{{d}^{2}}y}{d{{x}^{2}}}\]is also differentiable then its derivative is called third derivative of \[y\] which is denoted by \[\frac{{{d}^{3}}y}{d{{x}^{3}}}\]. Similarly \[{{n}^{th}}\] derivative of \[y\] is denoted by \[\frac{{{d}^{n}}y}{d{{x}^{n}}}\]. All these derivatives are called as successive derivatives and this process is known as successive differentiation. We also use the following symbols for the successive derivatives of \[g(x)\] :
\[{{y}_{1}},\,\,\,\,{{y}_{2}},\,\,\,\,{{y}_{3,}}.........,{{y}_{n}},......\] \[{y}',\,\,\,\,{y}'',\,\,\,\,{y}'''.........,{{y}^{n}},......\]
\[Dy,\,\,\,\,\,{{D}^{2}}y,\,\,\,\,{{D}^{3}}y.........,{{D}^{n}}y,......\], (where \[D=\frac{d}{dx}\])
\[\frac{dy}{dx},\,\,\,\,\frac{{{d}^{2}}y}{d{{x}^{2}}},\,\,\,\,\frac{{{d}^{3}}y}{d{{x}^{3}}},\,.......\,\,\,\frac{{{d}^{n}}y}{d{{x}^{n}}},...........\]
\[{f}'(x),\,\,\,\,{f}''(x),\,\,\,\,{f}'''(x),.........,{{f}^{n}}(x),......\]
If \[y=f(x)\], then the value of the \[{{n}^{th}}\] order derivative at \[x=a\] is usually denoted by \[{{\left( \frac{{{d}^{n}}y}{d{{x}^{n}}} \right)}_{x=a}}\] or \[{{({{y}_{n}})}_{x=a}}\] or \[{{({{y}^{n}})}_{x=a}}\] or \[{{f}^{n}}(a)\]
(2) \[{{n}^{th}}\] Derivatives of some standard functions :
(I) (a) \[\frac{{{d}^{n}}}{d{{x}^{n}}}\sin (ax+b)={{a}^{n}}\sin \left( \frac{n\pi }{2}+ax+b \right)\]
(b) \[\frac{{{d}^{n}}}{d{{x}^{n}}}\cos (ax+b)={{a}^{n}}\cos \left( \frac{n\pi }{2}+ax+b \right)\]
(II) \[\frac{{{d}^{n}}}{d{{x}^{n}}}{{(ax+b)}^{m}}=\frac{m\,!}{(m-n)\,!}{{a}^{n}}{{(ax+b)}^{m-n}},\] where \[m>n\]
Particular cases
(i) When \[m=n;\] \[{{D}^{n}}\{{{(ax+b)}^{n}}\}={{a}^{n}}.n\,!\]
(ii) (a) When \[a=1,b=0\], then \[y={{x}^{m}}\]
\[\therefore \]\[{{D}^{n}}({{x}^{m}})=m(m-1).......(m-n+1){{x}^{m-n}}=\frac{m!}{(m-n)!}{{x}^{m-n}}\]
(b) When \[m<n,\,{{D}^{n}}\{{{(ax+b)}^{m}}\}=0\]
(iii) When \[a=1,\,b=0\] and \[m=n\], then \[y={{x}^{n}};\,\therefore {{D}^{n}}({{x}^{n}})=n\,!\]
(iv) When \[m=-1,\,\,y=\frac{1}{(ax+b)}\]
\[{{D}^{n}}(y)={{a}^{n}}(-1)(-2)(-3)........(-n){{(ax+b)}^{-1-n}}\]
\[={{a}^{n}}{{(-1)}^{n}}(1.2.3......n){{(ax+b)}^{-1-n}}=\frac{{{a}^{n}}{{(-1)}^{n}}n\,!}{{{(ax+b)}^{n+1}}}\]
(III) \[\frac{{{d}^{n}}}{d{{x}^{n}}}\log (ax+b)=\frac{{{(-1)}^{n-1}}(n-1)!{{a}^{n}}}{{{(ax+b)}^{n}}}\]
(IV) \[\frac{{{d}^{n}}}{d{{x}^{n}}}({{e}^{ax}})={{a}^{n}}{{e}^{ax}}\]
(V) \[\frac{{{d}^{n}}({{a}^{x}})}{d{{x}^{n}}}={{a}^{x}}{{(\log a)}^{n}}\]
(VI) (i) \[\frac{{{d}^{n}}}{d{{x}^{n}}}{{e}^{ax}}\sin (bx+c)={{r}^{n}}{{e}^{ax}}\sin (bx+c+n\varphi )\]
where \[r=\sqrt{{{a}^{2}}+{{b}^{2}}};\,\,\varphi ={{\tan }^{-1}}\frac{b}{a}\]
(ii) \[\frac{{{d}^{n}}}{d{{x}^{n}}}{{e}^{ax}}\cos (bx+c)={{r}^{n}}{{e}^{ax}}\cos (bx+c+n\varphi )\]
