Calculus Derivative Product Rule with Sine, Cosine, & Powers. Fun Math Help Teaching Games by SliderMath Arithmetic Literacy.

Derivative Product Rule Practice with Sine Cosine & Power Rules     Math Help Fun     Game Tips:

- The derivative Product Rule is used on the product of two relations such as the product y=uv.
   In general, the Product Rule states that the derivative of the product function y=uv is    y'=uv'+u'v.

- For example, the unsimplified derivative of y=(x³x⁵) is    y'=(x³)(5x⁴)+(3x²)(x⁵).

- This game uses the Product Rule dy/dx = (u)(dv/dx) + (du/dx)(v) and also requires 4 other basic
   derivative rules: the Power Rule, the Sine Rule, the Cosine Rule, and the Constant times a Function Rule.
   Note that the symbols dy/dx, y' and f'(x) all represent the same derivative of y=f(x).

   1) The derivative Power Rule for y=xⁿ yields the derivative dy/dx = nx^n-1.
     For example, the derivatives of y=x⁴ and s(t)=t⁶ are respectively dy/dx=4x³ and s'(t)=6t⁵.

   2) The derivative Sine Rule for y=sin(x) yields the derivative dy/dx = cos(x).
     This means that the derivative of y=sin(x) is y'=cos(x).

   3) The derivative Cosine Rule for y=cos(x) yields the derivative dy/dx =-sin(x).
     This means that the derivative of y=cos(x) is y'=-sin(x).

   4) The derivative rule for a Constant times a Function such as y=cu yields the derivative dy/dx = c(du/dx).
      For example, the derivatives of y=8x⁴ and g(t)=5sin(t) are respectively y'=32x³ and g'(t)=5cos(t).

- Three more examples of the Product Rule y'=uv'+u'v follow:

   The derivative of y=(x³sin(x)) is    y'=(x³)(cos(x))+(3x²)(sin(x)).

   The unsimplified derivative of y=2x³[8cos(x)] is    y'=2x³[-8sin(x)]+6x²[8cos(x)].

   The derivative of y=(2sin(x))(8cos(x)) is    y'= -16sin(x)sin(x)+16cos(x)cos(x).

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- The Game score is reduced by the number of snowflake hits.
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