
Calculus Derivative Rules for Powers & Constants Math Help Fun Game Tips:  A derivative in Calculus can be described as an instantaneous rate of change. The slope of a curve at a point P is an example of a derivative. The speed of a moving particle at a time T is an example of a derivative.  A sample curve or motion can be represented by a relation such as y=x^{2} or the equivalent f(x)=x^{2}. The derivative corresponding to y=x^{2} can be represented as dy/dx = 2x or y' = 2x or f'(x) = 2x. The three derivative symbols dy/dx, y' and f'(x) are in common use by various authors.  Similarly, the relation represented as s=t^{3} or s(t)=t^{3} has it's corresponding derivative represented as ds/dt = 3t^{2} or s'(t) = 3t^{2}.  A derivative power rule is used to get dy/dx = 2x from the given relation y=x^{2}. The same derivative power rule is used to get ds/dt = 3t^{2} from the given relation s(t)=t^{3}.  The following 3 derivative rules are used in this game, with n representing a positive integer. 1) In general, the derivative power rule for y=x^{n} yields the derivative dy/dx = nx^{n1}. For example, the derivatives of y=x^{4} and s(t)=t^{6} are respectively dy/dx=4x^{3} and s'(t)=6t^{5}. 2) In general, the derivative constant rule for y=c yields the derivative dy/dx = 0. For example, the derivatives of y=3 and s(t)=8 are respectively dy/dx=0 and s'(t)=0. 3) The derivative rule for a constant times a power such as y=cx^{n} yields the derivative dy/dx = cnx^{n1}. For example, the derivatives of y=8x^{4} and g(t)=3t^{6} are respectively y'=32x^{3} and g'(t)=18t^{5}.  The score report automatically appears after you have made 10 choices.  The Math score is based on your choices only and does not count the butterfly hits.  The Game score is reduced by the number of butterfly hits.  The game can be played using the mouse by itself or using the keyboard by itself.  If the game doesn't respond to keyboard input, click inside the game area to reset the game's focus.  Adjust the game's speed by pressing the + or  key repeatedly.  Refresh/Reload the webpage to start over again. 
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