Question

how to prove that a function is differentiable?​

Answer 1

Answer:

example

Step-by-step explanation:

To show that f is differentiable at all x∈R, we must show that f′(x) exists at all x∈R.

Recall that f is differentiable at x if limh→0f(x+h)−f(x)h exists.

So for f(x)=−5x, we examine

limh→0−5(x+h)−(−5x)h=limh→0−5hh=limh→0−5=−5

And so we see that f is differentiable at all x∈R with derivative f′(x)=−5.

We could also say that if g(x) and h(x) are differentiable, then so too is f(x)=g(x)h(x) and that f′(x)=g′(x)h(x)+g(x)h′(x). Then let g(x)=x,h(x)=−5, noting that both are differentiable with derivates 1 and 0 respectively, leading to the same result.