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11.(vii)   Find the derivative of the following functions:  2 \tan x - 7 \sec x

Given  As we know  the property  Applying this property,

11(vi)   Find the derivative of the following functions: 5 \sin x - 6 \cos x + 7

Given, Now as we know the property  So, applying the property,

11.(v)   Find the derivative of the following functions:  3 \cot x + 5 \csc x

Given, As we know  the property  Applying the property,   Now As we know the quotient rule of derivative, So applying this rule, we get    

11.(iv)     Find the derivative of the following functions:  \csc x

Given : Now As we know the quotient rule of derivative, So applying this rule, we get

11 (iii)   Find the derivative of the following functions: 5 \sec x + 4 \cos x

Given As we know the property  Applying the property, we get

11(ii)   Find the derivative of the following functions:  \sec x

Given Now As we know the quotient rule of derivative, So applying this rule, we get

11.(i)   Find the derivative of the following functions: \sin x \cos x

Given, f(x)= Now, As we know the product rule of derivative, So, applying the rule here,

10.   Find the derivative of \cos x from first principle.

Given,  f(x)= Now, As we know, The derivative of any function at x is 

9(vi)   Find the derivative of \frac{2}{x+1}- \frac{x^2 }{3 x-1} 

Given As we know the quotient rule of derivative: and the property  So applying this rule, we get Hence 

9(v)   Find the derivative of x ^{-4} ( 3 - 4x ^{-5})

Given As we know, the property, and the property  applying that property we get

9.(iv)   Find the derivative of x ^5 ( 3 - 6 x ^{-9})

Given As we know, the property, and the property  applying that property we get

9(iii)   Find the derivative of x ^{-3} ( 5 + 3x ) 

Given As we know, the property, and the property  applying that property we get  

9.(ii)   Find the derivative of ( 5x^3 + 3x -1 ) ( x -1)

Given. As we know, the property, and the property  applying that property we get

 9.(i)   Find the derivative of  2x - 3/4

Given: As we know, the property, and the property  applying that property we get

8.   Find the derivative of  \frac{x ^n - a ^n }{x - a } for some constant a.

Given, Now As we know the quotient rule of derivative, So applying this rule, we get Hence

7.(iii)   For some constants a and b, find the derivative of \frac{x - a }{x -b }

Given, Now As we know the quotient rule of derivative, So applying this rule, we get Hence 

7.(ii)   For some constants a and b, find the derivative of ( ax ^2 + b)^2

Given  As we know, the property, and the property  applying those properties we get   

7.(i)  For some constants a and b, find the derivative of ( x - a ) ( x -b ) 

Given As we know, the property, and the property  applying that property we get   

6.   Find the derivative of x ^n + ax ^{n-1} + a ^ 2 x ^{n-2} + ....+ a ^ { n-1} x + a ^n for some fixed real number a.

Given  As we know, the property, applying that property we get   

5.   For the function f (x) = \frac{x^{100}}{100}+ \frac{x ^ {99}}{99} + ....+ \frac{x ^2 }{2}+ x+ 1 Prove that f '(1) =100 f '(0).

As we know, the property, applying that property we get  Now. So, Here   Hence Proved.
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