Question

The number of times the function $y=(x^2+1{)}^{80}$ should be differentiated to result in a polynomial degree $50$ is

• A. $70$
• B. $80$
• C. $110$
• D. $30$

Answer 1

The correct option is C $110$

y = ${(1+x^2)}^{80}$

Using Binomial expansion, we can write

y = $1+80x^2+\frac{80\times 79}{2!}{x}^4+...+x^{160}$

From here, we can see the highest coefficient of x is 160.

we know, $\frac{d\left(x^n\right)}{dx}=n{x}^{n-1}$

Thus one differentiation reduces the coefficient of $\left(x^n\right)$ to $x^{n-1}$.

So, in order to reduce $x^{160}$ to $x^{50}$, we need to differentiate the polynomial 160-50=110 times.

therefore the answer is OPTION C