Proof that change in a function can take a certain form $Δf(x)=g(x)⋅h+ε(h)h$ where h is the change in x and $Δf(x)$ is f(x+h)-f(x)?

Question

I was reading this page here: What are differentials?, and I noticed that the answer to the question based on re-writing the change of a function, f(x+h)-f(x). The change in f(x) was denoted as Δf(x,h) where h is the change in the input, x. I am confused on how the alternate form Δf(x,h)=k(x)⋅h+ε(h)h was derived, and if this form fits for all functions of x, such as exponential or trigonometric functions?

Thanks

score 1 · Answer 1 · answered Feb 08 '19 at 00:04

1

For fixed $x$ you can define $\epsilon (h)=\frac {f(x+h)-f(x)-g (x)h} h$. However you cannot make $\epsilon (h)$ independent of $x$ in general. For example, if $f(x)=e^{x}$ then $\epsilon (h)$ depends on $x$

answered Feb 08 '19 at 00:04

Kavi Rama Murthy

311,013

There is only one way of finding $\epsilon (h)$ for any function. Define it the way I have done in my answer. – Kavi Rama Murthy Feb 08 '19 at 00:33

Proof that change in a function can take a certain form $Δf(x)=g(x)⋅h+ε(h)h$ where h is the change in x and $Δf(x)$ is f(x+h)-f(x)?

1 Answers1