Theorem:For any constant   c   and any differentiable function   f,   d⁄dx(cf(x)) = c d⁄dx(f(x))  
Proof: This is just a matter of applying the definition of   d⁄dx   and then using the Constant Multiple Law for limits.

  1.Insert here    = climh → 0 1⁄h(f(x + h) - f(x))  

  2.Insert here Let   F(x) = cf(x). Then

  3.Insert here    = c d⁄dx(f(x))  

  4.Insert here    = limh → 0 1⁄h(cf(x + h) - cf(x))  

  5.Insert here   d⁄dx(F(x)) = limh → 0 1⁄h(F(x + h) - F(x))  

  6.Insert here    = limh → 0 c⁄h(f(x + h) - f(x))  

Time: 0 sec

Brief Instructions for solving this Text Puzzle: The puzzle pieces above have been shuffled, and your task is to move them around so that they end up in the correct position. Click a puzzle piece to select it, then click another puzzle piece to move (or swap) the selected piece to that particular position. When you think you are done, click on the Check button.

Text Puzzles by Peter Jipsen Chapman University