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Question
Determine a pattern for the nth derivative of f(x)=1/(6x+6), prove your pattern by finding the 4th derivative.
Note that x is always in the denominator for all of the derivatives, so we have a negative power on x. This would imply there is a -1^n in the nth derivative. Next note that the factor in front of x is 6. This would imply a 6^n in the nth derivative.
To find the fourth derivative, consider f(x)=(6x+6)^-1.
The first is -1((6x+6)^-2)*6 = -6(6x6)^-2.
To find the next, you would get -2*-6((6x+6)^-3)*6=72(6x+6)^-3. This is the second derivative.
Do the same operation again to get the third derivative, and then again to get to the fourth.
Also, not that I forgot to include an n! in the nth derivative do to the fact that the negative exponent is decreasing each time. This means that the absolute power of the integer value is increasing.
Also, note that the power would be one more than the derivative we're on. For ', the power is ^-2; for ", the power is ^-3, for ''', the power is -4.
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