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Example Find the derivative of the following function．

$\displaystyle \displaystyle{y = 11x^{5} - 6x^{3} + 8} $

Answer To find the derivative of polynomials, it is important to know the followings:

1. The derivative of a sum is the sum of derivatives． $(f(x) + g(x))' = f'(x) + g'(x)$  
2. The derivative of a constant multiple is the constant times the derivative． $(\alpha f(x))' = \alpha f'(x)$  

Apply these rules to the above problem, we have

$\displaystyle (11x^{5} - 6x^{3} + 8)'$ $\displaystyle =$ $\displaystyle (11x^5)' - (6x^3)' + 8'$  
  $\displaystyle =$ $\displaystyle 11(x^5)' -6(x^3)'$  
  $\displaystyle =$ $\displaystyle 55x^4 - 18x^2$