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If |x|<1 and y=1+x+x^2+....... oo , then...

If `|x|<1` and `y=1+x+x^2+....... oo` , then find the value of `(dy)/(dx)` .

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To solve the problem, we need to find the derivative of the function \( y = 1 + x + x^2 + \ldots \) (an infinite geometric series) with respect to \( x \), given that \( |x| < 1 \). ### Step-by-Step Solution: 1. **Identify the Infinite Series**: The series given is: \[ y = 1 + x + x^2 + x^3 + \ldots ...
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RD SHARMA-DIFFERENTIATION-Solved Examples And Exercises
  1. If y=sin^(-1)((1-x^2)/(1+x^2))+cos^(-1)((1-x^2)/(1+x^2)) , find (dy)/(...

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  2. If y=sec^(-1)((x+1)/(x-1))+sin^(-1)((x-1)/(x+1)),x > 0. Find (dy)/(dx)...

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  3. If |x|<1 and y=1+x+x^2+....... oo , then find the value of (dy)/(dx)...

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  4. If u=sin^(-1)((2x)/(1+x^2)) and v=tan^(-1)((2x)/(1-x^2)) , where -1 < ...

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  5. If f(x)=log{(u(x))/(v(x))},\ u(1)=v(1) and u^(prime)(1)=v^(prime)(1)=2...

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  6. If y=log(3x),\ \ x!=0 , find (dy)/(dx) .

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  7. If f(x) is an even function, then write whether f^(prime)(x) is eve...

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  8. If f(x) is an odd function, then write whether f^(prime)(x) is even...

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  9. Write the derivative of sinx with respect to cosx .

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  10. If f(x)=(log)(x^2)(logx) , then f^(prime)(x) at x=e is (a) 0 (b) 1 (c)...

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  11. The differential coefficient of f(logx) with respect to x , where f(...

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  12. The derivative of the function cot^(-1){(cos2x)^(1//2)} at x=pi//6 i...

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  13. Differential coefficient of sec(tan^(-1)x) is (a)x/(1+x^2) (b) xsqrt...

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  14. If y= tan^(-1)sqrt((1-sinx)/(1+sinx)), then the value of (dy)/(dx) at ...

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  15. If y=(1+1/x)^x , then (dy)/(dx)= (a)(1+1/x)^x{log(1+1/x)-1/(x+1)} (b)...

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  16. If x^y=e^(x-y) , then (dy)/(dx) is (a)(1+x)/(1+logx) (b) (1-logx)/(1+l...

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  17. Given f(x)=4x^8 , then (a)f'(1/2)=f'(-1/2) (b) f(1/2)=-f'(-1/2) (c) f...

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  18. If x=a\ cos^3theta,\ \ y=a\ sin^3theta , then (1+((dy)/(dx))^2)= (a)t...

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  19. If y=sin^(-1)((1-x^2)/(1+x^2)) , then (dy)/(dx)= (a)-2/(1+x^2) (b) 2/...

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  20. The derivative of sec^(-1)(1/(2x^2-1)) with respect to sqrt(1+3x) at x...

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