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# Problem with a Calculus 1 Derivative Question

Posted 9 years ago
 Let f '(x) = sin(x^2), Find h"(x) if h(x) = f(x^2)
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Posted 9 years ago
 Yes, but what did you try that didn't seem to work?
Posted 9 years ago
 I took the derivative of (sin(x^2)^2 and got 4xsin(x^2)cos(x^2) then i took the derivative of that answer but it was not correct. The correct answer is 2sin(x^4) + 8x^4 cos(x^4). I'm not understanding how to get this answer.
Posted 9 years ago
 I meant to say what did you type in what programming language that didn't work?
Posted 9 years ago
 Why don't you just type it in to get an idea? In:= DSolve[D[f[x], x] == Sin[x^2], f, x] Out= {{f -> Function[{x}, C + Sqrt[\[Pi]/2] FresnelS[Sqrt[2/\[Pi]] x]]}} In:= Clear[h, f] f[x_] := Sqrt[\[Pi]/2] FresnelS[Sqrt[2/\[Pi]] x] h[x_] := f[x^2] In:= D[h[x], x, x] Out= 8 x^4 Cos[x^4] + 2 Sin[x^4] So Mathematica agrees with the answer your book presents. That's good, but not good enough: For sure, if the solution is given, you are proposed to find the solution path: Try to find the solution without solving the differential equation. What happens if you don't know f itself, but only it's first derivative? In:= Clear[h, f] h[x] := f[x^2] In:= D[h[x], x, x] Out= 2 Derivative[f][x^2] + 4 x^2 (f^\[Prime]\[Prime])[x^2] What do you do? use the derivative in the first summand, take care about the dependency from x - easy, you get one summand of the final solution here take the derivative of the derivative (to get the second one), again take care about the dependency from x That's the way your teacher intends you to go; all steps can by done inside Mathematica. Go ahead!
Posted 9 years ago
 I took the derivative of (sin(x^2)^2 Oh no! That's the wrong beginning, sin(x^2)^2 is (f'(x))^2 and the exercise does neither require to consider it nor to take a derivation from it!