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Some interpretations of the derivative
Example Is f ( x ) = 2 x3 + 300 +4 increasing, decreasing or not changing at x = -2 ?
Solution: We already know that the rate of change of a function is specified by the functions derivative so all we have to do is it rewrite the function (to deal along with the second term) and then take the derivative.
f ( x ) = 2 x3 + 300 x-3+4 ⇒ f ′ ( x ) = 6x2 - 900x-4 = 6x2 - 900/x4
Note as well that we rewrote the last term in the derivative back as fraction. It is not something we've done to this point & it is only being done here to help with the evaluation in the next step. It's frequently easier to do the evaluation with +ve exponents.
Hence, upon evaluating derivative we get
f ′ ( -2) = 6 ( 4) - 900/16 = - 129/4 = -32.25
Hence, at x = -2 the derivative is negative and therefore the function is decreasing at x = -2 .
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