# JEE Main & Advanced Mathematics Differentiation Successive Differentiation or Higher Order Derivatives

## Successive Differentiation or Higher Order Derivatives

Category : JEE Main & Advanced

(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) $\frac{{{d}^{n}}}{d{{x}^{n}}}{{e}^{ax}}\cos (bx+c)={{r}^{n}}{{e}^{ax}}\cos (bx+c+n\varphi )$

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