Find an equation for the tangent line to the graph `y=6xcos10x`   a `x=pi`

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To determine the equation of the tangent line, its slope and point of tangency must be determined.

To solve for its slope, take the derivative of the given curve.

`y=6xcos(10x)`

`y'=6x*(cos(10x))' + cos(10x)*(6x)'`

`y'=6x*(-sin (10x)*10)+cos(10x)*6`

`y'=-60xsin(10x)+6cos(10x)`

Since m=y', plug-in x=pi to y'.

`m=y'=-60(pi)sin(10*pi)+6cos(10*pi)`

`m=0+6*1`

`m=6`

So, the slope of the tangent line is 6.

Next, to get the point of tangency, plug-in x=pi to y.

`y=6xcos(10x)=6*pi*cos(10*pi)=6pi*1`

`y=6pi`

Hence, the point of tangency is (pi, 6pi).

Now that the slope of the tangent line and its point of tangency are known, its equation can now be determined by using the point-slope form.

`y-y_1=m(x-x_1)`

`y-6pi=6(x-pi)`

`y-6pi=6x-6pi`

`y-6pi+6pi=6x-6pi+6pi`

`y=6x`

Therefore, the equation of the tangent line is `y=6x` .

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