(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
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