Stretch, compression, and reflection can change the derivative of a function while translation cannot. This does indeed work with all functions because stretch, compression, and reflection is multiplication and adding a negative, which show up in the derivative. Translation is just adding a constant, which doesn't show up. For example, the derivative of X^2 is 2x, while the derivative of 2x^2 is 4x. It changes. This however, does not work for composites of functions. The derivative of the example given at the point given is not the composition of the equations of the tangent lines. Therefore, compositions of functions break this rule.
To sum up, I feel this rule can be very useful in the future. By looking at the transformation of the function, I would no longer have to go through the difficult algebraic procedures and just predict the derivative based on the transformation of the function.
To sum up, I feel this rule can be very useful in the future. By looking at the transformation of the function, I would no longer have to go through the difficult algebraic procedures and just predict the derivative based on the transformation of the function.