# The Philosophy of Science Part 8: Differential Calculus

This video is part of our continuing series looking at the philosophical framework and intellectual tools that power the scientific method. One of the most important and powerful tools, which is foundational to the scientific method, is mathematics. We began developing this tool all the way back in the forth video of the series which started at the very beginning of mathematics with the simple enumeration of objects. From there we have progressed in our fifth and sixth videos developing more and more mathematical syntax including the concept of a mathematical function and also mathematical operations such as raising a number to a power. Once these mathematical operations were in place, we explored how these tools have been used to model the basic data we obtain from our universe when we observe objects in relative motion to one another including how an object’s motion is changed when an external force is applied. In our seventh video, we introduced the rather strange topic of the special theory of relativity which comes about because the laws of the universe must be conserved in every reference frame and so the propagation of electromagnetic radiation (light) must be constant independent of the relative motions of objects to one another. We showed for this experimental condition to be met, when translating space and time co-ordinates from one reference frame to another, a special factor must be applied which is called gamma and is equal to 1/√(1-v^{2}/c^{2}). In this video we now introduce the topic of differential calculus and develop the general derivative of a polynomial y=x^{n}. Calculus is a necessary tool required to prove Einstein’s famous equation Energy=mass × c^{2} where c is the speed of light.