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Composition of two differentiable functions

  • #1
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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
 

Answers and Replies

  • #2
Hurkyl
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Sure. You even know a formula for the derivative, right?
 
  • #3
Fredrik
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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 [itex]h\circ k[/itex] or [itex]x\mapsto h(k(x))[/itex]. (Note the special "mapsto" arrow).
 
  • #4
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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.
 

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