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 = nxn-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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