m1 = -(x - 1)/[f(x) - 3]
while that obtained by differentiation is
f '(x) = 2.08*x*sin(1.05*x) + 1.092*x^2*cos(1.05*x)
The functional dependence of these two curves is fortunately quite different, so that they intersect at the required value at a large angle. This enables an effective iterative procedure, similar to the Newton-Raphson method, to be employed. This converges to the required values in around 6-8 iterations, depending upon the initial estimate adopted. It requires that f(x) and f '(x) be calculated for each iteration, but otherwise the computational burden is relatively light.
The coordinates of A were obtained as (1.644756, 2.779180) to six-figure accuracy. As a check, the slope of AP and the gradient f '(x) were multiplied to check whether they were perpendicular, yielding a value for m1.f '(x) = -1.00002.