Composition of two differentiable functions

[michonamona]

Homework Statement

Is the composition of two differentiable functions always differentiable?

E.x.

h(x) = sin(x)
k(x) = 1/x for x not equal 0

Does this automatically mean h(k(x)) is differentiable?

Thank you,

M

 

[Hurkyl]

Sure. You even know a formula for the derivative, right?

 

[Fredrik]

I'll just comment about one of my little pet peeves. h(k(x)) is a number, not a function. The function you have in mind is written as $h\circ k$ or $x\mapsto h(k(x))$. (Note the special "mapsto" arrow).

 

[michonamona]

Thank you for your replies.

Sure. You even know a formula for the derivative, right?

So the composition of two differentiable functions is ALWAYS differentiable? I know the derivative of their composition, we just use the chain rule.

I'll just comment about one of my little pet peeves. h(k(x)) is a number, not a function. The function you have in mind is written as LaTeX Code: h\\circ k or LaTeX Code: x\\mapsto h(k(x)) . (Note the special "mapsto" arrow).

Thanks Fredrik. I never thought about that. Now I understand why they always use LaTeX Code: h\\circ k when referring to composition of functions.