The slope of the tangent to the curve y=sin(2x) at a given x value is found by calculating the derivative of the function, which is y'=2cos(2x). For y=cos(2x), the derivative is guessed to be -2sin(2x) using knowledge of basic derivatives and the chain rule.
To find the slope of the tangent to the curve y=sin(2x) at a given point using Desmos, you would create a table of x values and use Desmos to plot the function. Then, draw the tangent lines at those x values to find the slopes of each, using the formula for slope (rise over run). As the derivative of a function at a point gives us the slope of the tangent at that point, we'd find the derivative of y=sin(2x), which is y'=2cos(2x). This derivative function represents the slope of the tangent to the curve at any point x.
For the function y=cos(2x), an educated guess for its derivative can be made by knowing the derivatives of basic trigonometric functions. Since the derivative of cos(x) is -sin(x), by the chain rule, the derivative of cos(2x) would be -2sin(2x).