Derivatives of hyperbolic sine and cosine

Victor Hui
The orange coloured curve is the hyperbola (). Triangle PDQ is congruence to triangle PSR with PR equal to OP. Drag Q or P to approach each other and examine the congruence of the green and brown triangles. What is the orientational relationship between the segments OP and PR?
Differentiate the hyperbolic equation with respect to to yield . What is the geometric relationship between the vector and the gradient ? Prove that . What does the "unity" of the gradient tell us about the parameter ?