Find the derivative of the following function.

$3\cot x+5cosecx$

$\underset{x\to a}{\mathrm{l}\mathrm{i}\mathrm{m}}f\left(x\right)$ exists if and only if $\underset{x\to a^{-}}{\lim }f\left(x\right)=\underset{x\to a^{+}}{\lim }f\left(x\right)$

2. For a function $f\left(x\right)$ and a real number a, $\underset{x\to a}{\mathrm{l}\mathrm{i}\mathrm{m}}f\left(x\right)$ and $f\left(a\right)$ may not be same. In fact:

(i) $\underset{x\to a}{\mathrm{l}\mathrm{i}\mathrm{m}}f\left(x\right)$ exists but $f\left(a\right)$ (the value of $f\left(x\right)$ at $x=a$) may not exists.

(ii) The value $f\left(a\right)$ exists but $\underset{x\to a}{\mathrm{l}\mathrm{i}\mathrm{m}}f\left(x\right)$ does not exist.

(iii) $\underset{x\to a}{\mathrm{l}\mathrm{i}\mathrm{m}}f\left(x\right)$ and $f\left(a\right)$ both exist but are unequal.

(iv) $\underset{x\to a}{\mathrm{l}\mathrm{i}\mathrm{m}}f\left(x\right)$ and $f\left(a\right)$ both exist and are equal.

3. Algebra of Limits:

Let $\underset{x\to a}{\mathrm{l}\mathrm{i}\mathrm{m}}f\left(x\right)=l$ and $\underset{x\to a}{\lim }g\left(x\right)=m$. If $l$ and $m$ both exist, then

(i) $\underset{x\to a}{\lim }kf\left(x\right)=k\underset{x\to a}{\lim }f\left(x\right)=kl$

(ii) $\underset{x\to a}{\lim }\left(f\pm g\right)\mathrm{ }\left(x\right)=\underset{x\to a}{\lim }f\left(x\right)\pm \underset{x\to a}{\lim }g\left(x\right)=l\pm m$

(iii) $\underset{x\to a}{\lim }\left(fg\right)\left(x\right)=\underset{x\to a}{\lim }f\left(x\right)\underset{x\to a}{\lim }g\left(x\right)=lm$

(iv) $\underset{x\to a}{\lim }\left(\frac{f}{g}\right)\left(x\right)=\frac{\underset{x\to a}{\lim }f\left(x\right)}{\underset{x\to a}{\lim }g\left(x\right)}=\frac{l}{m}$

(v) $\underset{x\to a}{\lim }{\left(f\left(x\right)\right)}^{g\left(x\right)}={\left(\underset{x\to a}{\lim }f\left(x\right)\right)}^{\underset{x\to a}{\lim }g\left(x\right)}=l^m$

4. Following are some standard limits:

(i) $\underset{x\to a}{\lim }\frac{x^n-a^n}{x-a}=n{a}^{n-1}$

(ii) $\underset{x\to 0}{\lim }\frac{\sin x}{x}=1$

(iii) $\underset{x\to 0}{\lim }\frac{\tan x}{x}=1$

(iv) $\underset{x\to a}{\lim }\frac{\sin (x-a)}{x-a}=1$

(v) $\underset{x\to a}{\lim }\frac{\tan (x-a)}{x-a}=1$

(vi) $\underset{x\to 0}{\lim }\frac{\log (1+x)}{x}=1$

(vii) $\underset{x\to 0}{\lim }\frac{a^x-1}{x}={\log }_{e}a,a\ne 0,a>1$

(viii) $\underset{x\to 0}{\lim }\frac{e^x-1}{x}=1$

5. Sandwich Theorem:

Let $f,g,$ and $h$ be real functions such that $f\left(x\right)\le g\left(x\right)\le h\left(x\right)$ for all $x$ in the common domain of definition. For some real number $a$, if $\underset{x\to a}{\lim }f\left(x\right)=l=\underset{x\to a}{\lim }h\left(x\right),\text{ then }\underset{x\to a}{\lim }g\left(x\right)=l$.

6. Definition of Derivative:

(i) A function $f\left(x\right)$ is differentiable at $x=c$ if $\underset{x\to c}{\lim }\frac{f\left(x\right)-f\left(c\right)}{x-c}$ exists finitely.

This limit is called the derivative or differentiation of $f\left(x\right)$ at $x=c$ and is denoted by $f\text{'}\left(c\right)$.

(ii) The derivative of a function $f$ at $a$ is defined by $f^{\text{'}}\left(a\right)=\underset{h\to 0}{\lim }\frac{f\left(a+h\right)-f\left(a\right)}{h}$.

(iii) Mechanically, $\frac{d}{dx}\left(f\right(x\left)\right)$ measures the rate of change of $f\left(x\right)$ with respect to $x$.

7. Following are some standard derivatives:

(i) ddxxn=n xn-1

(ii) $\frac{d}{dx}\left(a^x\right)$$=a^x{\log }_{e}a; a>0, a\ne 1$

(iii) ddxex=ex

(iv) ddxloge x=1x

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

(vi) ddxcos x=-sin x

(vii) $\frac{d}{dx}\left(\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{ }x\right)=\mathrm{s}\mathrm{e}{\mathrm{c}}^{2}x$

(viii) $\frac{d}{dx}\left(\mathrm{c}\mathrm{o}\mathrm{t}x\right)=-\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{e}{\mathrm{c}}^{2}x$

(ix) $\frac{d}{dx}\left(\mathrm{s}\mathrm{e}\mathrm{c}x\right)=\mathrm{s}\mathrm{e}\mathrm{c}x \mathrm{t}\mathrm{a}\mathrm{n}x$

(x) $\frac{d}{dx}\left(\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{c}x\right)=-\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{c}x \mathrm{c}\mathrm{o}\mathrm{t}x$

(xi) Differentiation of a constant function is zero i.e., ddx(c)=0.

8. Algebra of derivative of functions:

If $f\left(x\right)$ and $g\left(x\right)$ are differentiable functions, then

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

(ii) Product Rule:

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

(iii) Quotient Rule:

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

(iv) $\frac{d}{dx}\left(kf\left(x\right)\right)=k\frac{d}{dx}\left(f\left(x\right)\right)$, where $k$ is a constant function.