Prove that f(x) is differentiable U ( x ) = x 2 + f ( x ) is differentiable for all xHere's my proof please let me know if it's OK: x 2 is differentiable for all x because it's a polynomial. Thus if f(x) is differentiable for all x then U(x) is differentiable for all x. Let U(x) be differentiable for all x. Thus according to addition arithmetic U ( x ) − x 2 is differentiable for all x.But U ( x ) − x 2 = x 2 + f ( x ) − x 2 = f ( x ). Thus the addition that differentiable itself equals f(x) therefore f(x) is differentiable as well.I have to admit that there was a very similar exercise in my class and from which I used the "addition arithmetic" however I'm not sure what it would refer to. Does this refer to the fact that according to derivatives arithmetic the result of addition of 2 function which are differentiable themselves is differentiable? But we didn't actually derive these functions.

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Prove that f(x) is differentiable   U  (  x  )  =      x    2    +  f  (  x  ) is differentiable for all xHere&#039;s my proof please let me know if it&#039;s OK:      x    2   is differentiable for all x because it&#039;s a polynomial. Thus if f(x) is differentiable for all x then U(x) is differentiable for all x. Let U(x) be differentiable for all x. Thus according to addition arithmetic   U  (  x  )  −      x    2   is differentiable for all x.But   U  (  x  )  −      x    2    =      x    2    +  f  (  x  )  −      x    2    =  f  (  x  ). Thus the addition that differentiable itself equals f(x) therefore f(x) is differentiable as well.I have to admit that there was a very similar exercise in my class and from which I used the &quot;addition arithmetic&quot; however I&#039;m not sure what it would refer to. Does this refer to the fact that according to derivatives arithmetic the result of addition of 2 function which are differentiable themselves is differentiable? But we didn&#039;t actually derive these functions.

teleriasacr · Accepted Answer

Step 1If you are indeed allowed to use the fact that the sum of differentiable functions is again differentiable and that polynomials are differentiable as well, then your work is fine.Step 2U(x) is the sum of two differentiable functions, hence differentiable. And if U(x) is differentiable, then so is   U  (  x  )  +  (  −      x    2    )  =  f  (  x  )

Prove that f(x) is differentiable iff U(x)=x^2+f(x) is differentiable for all x

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