Left Hand Derivative

Q. Find the derivative of the following function:
f(x) = $\frac{cosx}{1+sinx}$

Q. Find the derivative of the following function:
f(x) = $(x^2+1)cosx$

Q. Find the derivative of the following function:
f(x)= $si{n}^{n}x$

Q. Find the derivative of the following function:
f(x)= $\frac{a}{x^4}-\frac{b}{x^2}+cosx$

Q.

Find the derivative of $(9x^2+\frac{3}{x}+5sinx)$ with respect to x.

Q. Find the derivative of the following function:
f(x)= $\frac{sin(x+a)}{cosx}$

Q. Find the derivative of the following function:
f(x)= $\frac{4x+5sinx}{3x+7cosx}$

Q. The left-hand derivative of $f\left(x\right)=\left[x\right]sin\left(\pi x\right)$ at x= k, k is an integer and [x] = greatest integer, is

1. $(-1{)}^{k-1}(k-1)\pi$
2. $(-1{)}^k(k-1)\pi$
3. $(-1{)}^{k}k\pi$
4. $(-1{)}^{k-1}k\pi$

Q.

$\text{Derivatie of}{e}^{si{n}^{-1}x}w.r.t.si{n}^{-1}is$

1. 

2. 

3. 

4. 

Q.

(i) Evaluate ${lim}_{x\to 1}\frac{x+x^2+x^3+...+x^n-n}{x-1}$

(ii) Find the derivative $\text{sqrt{sin x}}$ from first principle.

Q. Find the derivative of the following function:
f(x) = $\frac{x^{2}cos\left(\frac{\pi }{4}\right)}{sinx}$

Q. Find the derivative of the following function:
f(x)= $\frac{a+bsinx}{c+dcosx}$

Q. The left-hand derivative of $f\left(x\right)=\left[x\right]\sin \left(\pi x\right)$ at $x=k$
where $k$ is an integer and $\left[x\right]$ is the greatest integer function, is

1. $(-1{)}^{k}k\pi$
2. $(-1{)}^k(k-1)\pi$
3. $(-1{)}^{k-1}k\pi$
4. $(-1{)}^{k-1}(k-1)\pi$

Q. The left-hand derivative of f(x) = [x] sin ($\pi$ x) at x = k, k is an integer and [x] = greatest integer $\le$ x, is

[IIT Screening 2001]

1. $(-1{)}^{k-1}(k-1)\pi$
2. $(-1{)}^k(k-1)\pi$
3. $(-1{)}^{k}k\pi$
4. $(-1{)}^{k-1}k\pi$

Q. The left-hand derivative of $f\left(x\right)=\left[x\right]\sin \pi x$ at $x=k,k$ is an integer is

1. $(-1{)}^{k-1}(k-1)\pi$
2. $(-1{)}^{k}k\pi$
3. $(-1{)}^{k-1}k\pi$
4. $(-1{)}^k(k-1)\pi$

Q.

\(lim_{x\to 0}\) \(\frac{x({e}^{x}-1)}{1-cos x}\) =