`f(x)=e^(x)-e^(-x)-2 sin x -(2)/(3)x^(3).` Then the least value of n for which `(d^(n))/(dx^(n))f(x)|underset(x=0)` is nonzero is

Options

A. 5
B. 6
C. 7
D. 8

Explanation

To find the least value of \( n \) for which \( \frac{d^n}{dx^n} f(x) \bigg|_{x=0} \) is nonzero, we will differentiate the function \( f(x) \) step by step and evaluate the derivatives at \( x = 0 \).

Given:
\[ f(x) = e^x - e^{-x} - 2 \sin x - \frac{2}{3} x^3 \]

### Step 1: Calculate the first derivative \( f'(x) \)
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