If `(1+x)^n=c_0+C_1x+C_2x^2++C_n x^n `,using derivative prove that `C_1+2C_2++nC_n=n .2^(n-1)` `C_1-2C_2+3C_3++(-1)^(n-1)nC_n=0`

Explanation

We have, `(1+x)^{n}=C_{0}+C_{1} x+C_{2} x^{2}+ldots+C_{n} x^{n}`


and `(1-x)^{n}=C_{0}-C_{1} x+C_{2} x^{2}-C_{3} x^{3}+ldots+(-1)^{n} C_{n} x^{n}`


Put `x=y^{2}`



`(1-y^{2})^{n}=C_{0}-C_{1} y^{2}+C_{2} y^{4}-C_{3} y^{6}+ldots+(-1)^{n} C_{n} y^{2 n}`



Multiply both sides by `y^{3}`, we get



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