Clamor derivitives
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y=x^42-32x^2-x y' = 42x^41 - 32(2)x - 1 y' = 42x^41 -64x-1
To find the ARITHMETIC mean of 4 and 10, you add them up and then divide by n number of values: (4+10)/2 = 7 To find the GEOMETRIC mean, you multiply 4 and 10, and then find the nth root: the square root of 40 is 6.32 (to 3 significant figures).
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Smooth function has derivatives of all orders. Polynomials have derivatives of all orders, thus polynomials are smooth functions. For example: f(x)=2x+3 => f'(x)=2 => f''(x)=0 => f'''(x)=0... So all derivatives exist. (Derivative being zero is ok.) Their continuity can be proven using the Weierstrass (epsilon-delta) definition.