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30.   Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

$\frac{x }{\sin ^ n x }$

Given, Now, As we know the derivative of any function     Also chain rule of derivative, Hence the derivative of the given function is

29. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): $( x + \sec x ) ( x - \tan x )$

Given  Now, As we know  the Multiplication property of derivative, Also the property  Applying those properties we get,  the derivative of the given function is,

28.   Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

$\frac{x }{ 1+ \tan x }$

Given  Now, As we know the derivative of any function

27.  Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

$\frac{x ^2 \cos ( \pi /4 )}{\sin x }$

Given, Now, As we know the derivative of any function   Now, As we know the derivative of any function   Hence the derivative of the given function is

26.   Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

$\frac{4x + 5 \sin x }{3x+ 7 \cos x }$

Given, Now, As we know the derivative of any function   Also the property  Applying those properties,we get

25.   Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): $( x+ \cos x ) ( x - \tan x )$

Given, And the Multiplication property of derivative, Also the property  Applying those properties we get,

24.   Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):  $( ax ^2 + \sin x ) ( p + q \cos x )$

Given, Now As we know the Multiplication property of derivative,(the product rule) And also the property Applying those properties we get,

23.  Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): $( x^2 +1 ) \cos x$

Given  Now, As we know the product rule  of derivative, The derivative of the given function is

22.  Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): $x ^ 4 ( 5 \sin x - 3 \cos x )$

Given  Now, As we know, the Multiplication property of derivative, Hence derivative of the given function is:

21.   Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

$\frac{\sin ( x+a )}{ \cos x }$

Given,

$f(x)=\frac{\sin ( x+a )}{ \cos x }$

Now, As we know the derivative of any function

$\frac{d(\frac{y_1}{y_2})}{dx}=\frac{y_2d(\frac{dy_1}{dx})-y_1(\frac{dy_2}{dx})}{y_2^2}$

Hence the derivative of the given function is:

$\frac{d(\frac{\sin(x+a)}{\cos x})}{dx}=\frac{(\cos x)(\frac{d(\sin (x+a))}{dx})-\sin(x+a)(\frac{d(\cos x)}{dx})}{(\cos x)^2}$

$\frac{d(\frac{\sin(x+a)}{\cos x})}{dx}=\frac{(\cos x)(\cos(x+a))-\sin(x+a)(-\sin (x))}{(\cos x)^2}$

$\frac{d(\frac{\sin(x+a)}{\cos x})}{dx}=\frac{(\cos x)(\cos(x+a))+\sin(x+a)(\sin (x))}{(\cos x)^2}$

$\frac{d(\frac{\sin(x+a)}{\cos x})}{dx}=\frac{\cos (x+a-x)}{(\cos x)^2}$

$\frac{d(\frac{\sin(x+a)}{\cos x})}{dx}=\frac{\cos (a)}{(\cos x)^2}$

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20.  Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

$\frac{a + b \sin x }{c+ d \cos x }$

Given Function

$f(x)=\frac{a + b \sin x }{c+ d \cos x }$

Now, As we know the derivative of any function  of this type is:

$\frac{d(\frac{y_1}{y_2})}{dx}=\frac{y_2d(\frac{dy_1}{dx})-y_1(\frac{dy_2}{dx})}{y_2^2}$

Hence derivative of the given function will be:

$\frac{d(\frac{a+b\sin x}{c+d\cos x})}{dx}=\frac{(c+d\cos x)(\frac{d(a+b\sin x)}{dx})-(a+b\sin x)(\frac{d(c+d\cos x x)}{dx})}{(c+d\cos x)^2}$

$\frac{d(\frac{a+b\sin x}{c+d\cos x})}{dx}=\frac{(c+d\cos x)(b\cos x)-(a+b\sin x)(d(-\sin x))}{(c+d\cos x)^2}$

$\frac{d(\frac{a+b\sin x}{c+d\cos x})}{dx}=\frac{cb\cos x+bd\cos^2 x+ad\sin x+bd\sin^2 x}{(c+d\cos x)^2}$

$\frac{d(\frac{a+b\sin x}{c+d\cos x})}{dx}=\frac{cb\cos x+ad\sin x+bd(\sin^2 x+\cos^2 x)}{(c+d\cos x)^2}$

$\frac{d(\frac{a+b\sin x}{c+d\cos x})}{dx}=\frac{cb\cos x+ad\sin x+bd}{(c+d\cos x)^2}$

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19.  Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): $\sin^ n x$

Given, Now, As we know the chain rule of derivative, And, the property, Applying those properties, we get Hence Derivative of the given function is

18.  Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

$\frac{\sec x -1}{\sec x +1}$

Given, which also can be written as  Now, As we know the derivative of such function   So, The derivative of the function is, Which can also be written as  .

17.  Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): $\frac{\sin x + \cos x }{\sin x - \cos x }$

Given  Also can be written as which further can be written as Now,

16.   Find the derivative of the following functions (it is to be understood that a, b, c, d,p, q, r and s are fixed non-zero constants and m and n are integers):

$\frac{\cos x }{1+ \sin x }$

Given, Now, As we know the derivative of any function   Hence, The derivative of f(x) is  (Answer)

15.  Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):  $\csc x \cot x$

Given,  the Multiplication property of derivative, Applying the property  Hence derivative of the function is .

14.  Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): $\sin ( x + a )$

Given, Now, As we know the chain rule of derivative, Applying this property we get,

13  Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):  $( ax + b ) ^ n ( cx + d ) ^ m$

Given Now, As we know the chain rule of derivative, And the Multiplication property of derivative, And, the property, Also the property  Applying those properties we get,

12.  Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

$( ax + b ) ^ n$

Given Now, As we know the chain rule of derivative, And, the property, Also the property  applying those properties we get,

11  Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):  $4 \sqrt x - 2$

Given It can also be written as  Now, As we know, the property, and the property  applying that property we get
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