# Differentiation and simplification of composite function

in an introductory text on calculus for economists I found the following $$y=f(\bar{h}-t)$$ then, differentiating $y$ in respect to $t$ $$\frac{dy}{dt}=f'(\bar{h}-t)\frac{d}{dt}(\bar{h}-t)$$ then simplifying $$\frac{dy}{dt}=-f'(\bar{h}-t)$$ While I think I understood the chain rule-based differentiation (derivative of outer function times inner function times derivative of inner function), I struggle to work out the simplification. Can please anyone be so patient to help with the simplification? Thank you

• This is correct. There is not a simplification, it just computed the derivative and treated $\overline y$ as a constant nothing more. WHich i dont know can be true :). – Brethlosze Jun 4 '17 at 15:54
• @hyprfrco You mean $\overline {h}$ – hamam_Abdallah Jun 4 '17 at 16:07
• yes!.. my mistake – Brethlosze Jun 4 '17 at 16:18
• Thank you very much. Now it makes sense, I got confused because the author said he was doing simplification. – Nicola Pensiero Jun 4 '17 at 18:06

## 1 Answer

As you said, first you take the outer derivative, resulting in the prime notation. Then you take the inner derivative, which is equal to -1. Multiplying then together gives the result. It isn't simplified, just a slightly odd notation I think. Obviously, all this is assuming h is not a function of t.