To find the derivative of f(x) at x = 1, we can use the formula for implicit differentiation:
d/dx[f(x) + x^2[f(x)]^5] = 0
d/dx[f(1) + 1^2[f(1)]^5] = 0
d/dx[2 + 1[2]^5] = 0
d/dx[2 + 1[2]^5] = 0
d/dx[2 + 32] = 0
0 + 32 = 0
Therefore, the derivative of f(x) at x = 1 is 32.
To find the equation of the tangent line to the curve at the given point (pi/2, pi/4), we can use the formula for the slope of a tangent line:
m = f'(x) = d/dx[y sin 12x] / d/dx[x cos 2y]
m = d/dx[pi/4 sin 12(pi/2)] / d/dx[pi/2 cos 2(pi/4)]
m = d/dx[pi/4 sin 6pi] / d/dx[pi/2 cos pi/2]
m = 0 / 0
Since the derivative of the numerator is 0 and the derivative of the denominator is 0, the slope of the tangent line is undefined. This means that the tangent line is vertical and parallel to the y-axis.
The equation of the tangent line at the given point is therefore of the form:
x = c
where c is the x-coordinate of the given point, which is pi/2. Therefore, the equation of the tangent line is:
x = pi/2
Note that this is just the equation of a vertical line with an x-intercept at pi/2.
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