Question

How do you find f'(x) using the definition of a derivative for `f(x)=(1-6t)/(5+t)`?

Answer 1

`-31/(5+t)^2`

Explanation:

`f(t)=(1-6t)/(5+t)`
`f(t+h)=(1+6(t+h))/(5+(t+h))`
`f(t+h)-f(t)=(1-6(t+h))/(5+(t+h))-(1-6t)/(5+t)=(5+t-6(t^2+5h+5t+ht)-5-t-h+30t+6t^2+6ht)/((5+t+h)(5+t))=(-31h)/((5+t+h)(5+t))`
`Lt_(h->0)((f(t+h)-f(h))/h)=Lt_(h->0)(-31h)/(h(5+t+h)(5+t))=Lt_(h->0)(-31)/((5+t+h)(5+t))=-31/(5+t)^2`