Answer :
The equation of the tangent line to the curve is:
[tex]y - 4 = 0.0024(x - 2)[/tex]
Which has a slope of 0.0024.
The tangent line of a curve [tex]f(x)[/tex] at a point [tex](x_0,y_0)[/tex] is given by:
[tex]y - y_0 = f^{\prime}(x_0)(x - x_0)[/tex]
In this problem, the curve is:
[tex]f(x) = e^{-4x}\sin{6x}[/tex]
The derivative is:
[tex]f^{\prime}(x) = -4e^{-4x}\sin{6x} + 6e^{-4x}\cos{6x}[/tex]
Point P(2,4), thus [tex]x_0 = 2, y_0 = 4[/tex].
The slope is:
[tex]f^{\prime}(2) = -4e^{-4(2)}\sin{6(2)} + 6e^{-4(2)}\cos{6(2)} = 0.0024[/tex]
Then, the equation of the line is:
[tex]y - y_0 = f^{\prime}(x_0)(x - x_0)[/tex]
[tex]y - 4 = 0.0024(x - 2)[/tex]
A similar problem is given at https://brainly.com/question/22426360