Differentiate w.r.t. $x$ the function
$x^{x^2-3}+{\left(x-3\right)}^{x^2},$ for $x>3.$

1. Continuity:

(i) A real valued function $f\left(x\right)$ is continuous at a point $a$ in its domain if $\underset{x\to a}{\mathrm{l}\mathrm{i}\mathrm{m}}f\left(x\right)=f\left(a\right)$.  i.e. the limit of the function at $x=a$ is equal to the value of the function at $x=a.$

(ii) A function $f\left(x\right)$ is said to be continuous if it is continuous at every point on its domain.

(iii) Sum, difference, product and quotient of continuous functions are continuous i.e., if $f\left(x\right)$ and $g\left(x\right)$ are continuous functions on their common domain, then $f\pm g, fg,\frac{f}{g}\left(g\ne 0\right), kf$ ($k$ is a constant) are continuous.

(iv) Let $f$ and $g$ be real functions such that $fog$ is defined. If $g$ is continuous at $x=a$ and $f$ is continuous at $g\left(a\right)$ then $fog$ is continuous at $x=a.$

(v) Following functions are everywhere continuous:

(a) A constant function

(b) The identity function

(c) A polynomial function

(d) Modulus function

(e) Exponential function

(f) Sine and Cosine functions

(vi) Following functions are continuous in their domains:

(a) A logarithmic function

(b) A rational function

(c) Tangent, cotangent, secant and cosecant functions.

(d) All inverse trigonometric functions are continuous in their respective domains.

2. Differentiability:

(i) A real valued function $f\left(x\right)$ defined on $(a,b)$ is said to be differentiable at $x=c\in \left(a,b\right)$ if limx→cf(x)-f(c)x-cexists finitely.

$\mathrm{LHD}\text{ at }\left(x=c\right)=\underset{h\to 0}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{f(c-h)-f\left(c\right)}{-h}$ and $\mathrm{RHD} at \left(x=c\right)=\underset{h\to 0}{\lim }\frac{f(c+h)-f\left(c\right)}{h}$

(ii) A function is said to be differentiable, if it is differentiable at every point in its domain.

(iii) Every differentiable function is continuous but, the converse is not necessarily true.

(iv) The sum, difference, product and quotient of two differentiable functions is differentiable.

(v) If a function $f\left(x\right)$ is differentiable at every point in its domain, then limh→0f(x+h)-f(x)hor limh→0f(x-h)-f(x)-h is called the derivative or differentiation of $f$ at $x$ and is denoted by $f\text{'}\left(x\right)$ or $\frac{d}{dx}\left(f\left(x\right)\right)$.

(i) $\frac{d}{dx}\left\{cf\left(x\right)\right\}=c\frac{d}{dx}\left\{f\left(x\right)\right\}$

(ii) $\frac{d}{dx}\left\{f\left(x\right)\pm g\left(x\right)\right\}=\frac{d}{dx}\left\{f\left(x\right)\right\}\pm \frac{d}{dx}\left\{g\left(x\right)\right\}$

(iii) $\frac{d}{dx}\left\{f\left(x\right)g\left(x\right)\right\}=f\left(x\right)\frac{d}{dx}\left\{g\left(x\right)\right\}+g\left(x\right)\frac{d}{dx}\left\{f\left(x\right)\right\}$

(iv) $\frac{d}{dx}\left\{\frac{f\left(x\right)}{g\left(x\right)}\right\}=\frac{g\left(x\right)\frac{d}{dx}\left\{f\left(x\right)\right\}-f\left(x\right)\frac{d}{dx}\left\{g\left(x\right)\right\}}{{\left\{g\left(x\right)\right\}}^2}$

4. Derivatives of some standard functions:

(i) $\frac{d}{dx}\left(x^n\right)=n{x}^{n-1}$

(ii) ddxex=ex

(iii) ddxax=axlogea

(iv) ddxlogex=1x

(v) ddxlogax=1xlogea

5. Derivatives of trigonometric functions:

(i) $\frac{d}{dx}\left(\sin x\right)=\mathrm{c}\mathrm{o}\mathrm{s}x$

(ii) $\frac{d}{dx}\left(\cos x\right)=-\mathrm{s}\mathrm{i}\mathrm{n}x$

(iii) $\frac{d}{dx}\left(\tan x\right)=\mathrm{s}\mathrm{e}{\mathrm{c}}^{2}x$

(iv) $\frac{d}{dx}\left(\sec x\right)=\sec x \mathrm{t}\mathrm{a}\mathrm{n}x$

(v) $\frac{d}{dx}\left(\cot x\right)=-{\mathrm{cosec}}^{2}x$

(vi) $\frac{d}{dx}\left(\mathrm{cosec}x\right)=-\mathrm{cosec}x\cot x$

6. Derivatives of inverse trigonometric functions:

(i) $\frac{d}{dx}\left(\mathrm{s}\mathrm{i}{\mathrm{n}}^{-1}x\right)=\frac{1}{\sqrt{1-x^2}},-1<x<1$

(ii) ddxcos-1x=-11-x2,-1<x<1

(iii) ddxtan-1x=11+x2,-∞<x<∞

(iv) ddxcot-1x=-11+x2,-∞<x<∞

(v) $\frac{d}{dx}\left(\mathrm{s}\mathrm{e}{\mathrm{c}}^{-1}x\right)=\frac{1}{\left|x\right|\sqrt{x^2-1}},\left|x\right|>1$

(vi) $\frac{d}{dx}\left({\cos }^{-1}x\right)=\frac{-1}{\left|x\right|\sqrt{x^2-1}},\left|x\right|>1$

7. Logarithmic differentiation: 

If $y={\left\{f\left(x\right)\right\}}^{\left\{g\left(x\right)\right\}}$ then $\frac{d}{dx}\left[\{f\left(x\right){\}}^{\left\{g\left(x\right)\right\}}\right]={\left\{f\left(x\right)\right\}}^{g\left(x\right)}\left\{\frac{g\left(x\right)}{f\left(x\right)}\frac{d}{dx}\left\{f\left(x\right)\right\}+\mathrm{l}\mathrm{o}\mathrm{g}f\left(x\right)\frac{d}{dx}\left\{g\left(x\right)\right\}\right\}$

8. Derivatives of Functions in Parametric Forms: 

If $x=\varphi \left(t\right)$ and $y=\mathrm{\psi }\left(t\right)$ then $\frac{dy}{dx}=\frac{{\displaystyle \frac{dy}{dt}}}{{\displaystyle \frac{dx}{dt}}}$.

9. Derivatives of Determinants:

If $f\left(x\right), g\left(x\right), u\left(x\right)$ and $v\left(x\right)$ are function of $x$ and $∆$ is a determinant given by $\mathrm{\Delta }\left(x\right)=\left|\begin{array}{ll}f\left(x\right)& g\left(x\right)\\ u\left(x\right)& v\left(x\right)\end{array}\right|$ Then, $\frac{d}{dx}\left\{\mathrm{\Delta }\left(\mathrm{x}\right)\right\}=\left|\begin{array}{ll}{f}^{\mathrm{\text{'}}}\left(x\right)& g^{\mathrm{\text{'}}}\left(x\right)\\ u\left(x\right)& v\left(x\right)\end{array}\right|+\left|\begin{array}{ll}f\left(x\right)& g\left(x\right)\\ u^{\mathrm{\text{'}}}\left(x\right)& v^{\mathrm{\text{'}}}\left(x\right)\end{array}\right|$ Also, $\frac{d}{dx}\left\{\mathrm{\Delta }\left(\mathrm{x}\right)\right\}=\left|\begin{array}{ll}{f}^{\mathrm{\text{'}}}\left(x\right)& g\left(x\right)\\ u^{\mathrm{\text{'}}}\left(x\right)& v\left(x\right)\end{array}\right|+\left|\begin{array}{ll}f\left(x\right)& g^{\mathrm{\text{'}}}\left(x\right)\\ u\left(x\right)& v^{\mathrm{\text{'}}}\left(x\right)\end{array}\right|$

10. Second Order Derivatives:

(i) If $y=f\left(x\right),$ then ddxdydx is called second order derivative of $y$ with respect to $x$ and is denoted by d2ydx2 or $y_2$ or $y".$ 

(ii) If $x=f\left(t\right)$ and $y=g\left(t\right),$ then d2ydx2=ddxg'tf't or $\frac{d^{2}y}{d{x}^2}=\frac{d}{dt}\left\{\frac{g\text{'}\left(t\right)}{f\text{'}\left(t\right)}\right\}\times \frac{dt}{dx}$ or d2ydx2=f'tg"t-g'tf"tf't

11. Rolle’s Theorem:

Let $f$ be a real value of function defined on the closed interval $\left[a, b\right]$ such that

(i) It is continuous on $\left[a,b\right]$

(ii) It is differentiable on $\left(a,b\right)$ and

(iii) $f\left(a\right)=f\left(b\right)$

Then, there exists at least one real number $c\in \left(a,b\right)$ such that $f\text{'}\left(c\right)=0$.

12. Lagrange's Mean Value Theorem:

Let $f\left(x\right)$ be a function defined on $[a, b]$ such that

(i) It is continuous on $[a,b]$ and

(ii) Differentiable on $(a,b)$

Then, there exists at least one $c\in (a,b)$ such that $f\text{'}\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$