How do I find f'(x) of f(t)=9/(t^3)?

Question

I do not understand the power rules. Please include a detailed step by step explanation so I can follow along and do similar problems myself. I'm really lost on this unit! Thank you!

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Answer 1 · 2018-02-24T01:29:57

#-27/t^4#

#d/dt(9/t^3)=9d/dt(1/t^3)#

We can always factor out constants when differentiating. It often cleans up the actual differentiation process.

Recall that #1/x^a=x^-a#

This means that

#1/t^3=t^-3#

Writing in this form allows for actually using the power rule.

The power rule states that

#d/dxx^n=nx^(n-1)#

In other words, the derivative of a term raised to the #n^(th)# power is #n# multiplied by that same term raised to the #(n-1)^(th)# power.

For our expression, #t^-3#, we can see #n=-3#

If #n=-3#, #(n-1)=-3-1=-4#

So,

#9d/dtt^-3=(9)(-3)t^-4=-27t^-4=-27(t^-4)=(-27)(1/t^4)=-27/t^4#