Visual for derivative of tangent.

For compactness sake, I have notated the area of the blue triangle with a capital A.

When tan(t), which I have written above, is negative, imagine the blue triangle's area, A, being negative. The orange triangle's area, however, remains positive, even when sec(t) is negative, due to the formula for the area of the orange triangle containing a squaring operation.

Yes, the orange triangle's area is the derivative of the blue triangle's area with respect to t. If you squish the orange triangle parallel to d◈/dt, towards the blue triangle, this becomes clearer.