How do you find the derivative of a root. such as `root(4)(t^5)` or `root(5)(t)+root(4)(t^5)`

Expert Answers
tjbrewer eNotes educator| Certified Educator

Mathematical roots can also be expressed as exponents.  Your first expression `root(4)(t^5)` is the same as `t^(5/4)` I'm sure that you can differentiate that expression (i.e. find it's derivative).  According to the calculus I remember `dy/dt t^(5/4) = 1 1/4 t^(1/4) = 1 1/4 root(4)(t)` Hope this helps. 

oldnick | Student

`d/dt root(4)(t^5)= d/dt t^(5/4)=5/4 t^(5/4-1))=5/4t^(1/4)=5/4root(4)(t)` 

in thre seond case, tha derivatives of sum of funciìtion is sum of derivatives o f the functions:

`d/dt [f(t)+g(t)]=d/dt f(t)+d/dt g(t)`


`d/dt[root(5)(t)+root(4)(t^5)]=d/dt root(5)(t) + d/dt root(4)(t^5)]=d/dt t^(1/5) + d/dt t^(5/4)=```

`=1/5 t^(1/5-1)+ 5/4 t ^(5/4-1)` `=1/5 t^(-4/5)+ 5/4 t^(1/4)=` `1/5 1/t^(4/5) +5/4 t^(1/4)=`

`=1/5 1/root(5)(t^4) +5/4 root(4)(t)` `=1/5 1/root(5)(t^4) root(5)(t)/root(5)(t)+ 5/4 root(4)(t)` `=1/5 root(5)(t)/t+5/4 root(4)(t)`