...Good day Newton, These kinds of proofs make mathematics personally so much fun to study. Great presentation, and I just wanted to pass on to you that I learned that the procedure is called "First PrincipleS" with an "S"... Thank you and take care, Jan-W
There is another way to prove this from first principles. Your approach is much simpler and quicker but the difference quotient: [tan(x+h) - tan(x)]/h can be expressed as [tan(h)(1 + tan^2(x))/(h(1-tan(h)tan(x)))]. Taking the limit of this expression as h tends toward 0 yields 1 + tan^2(x), which is the Pythagorean identity for sec^2(x) ◼
Yes, using the angle sum identity for tangent is super straight-forward. Just put the terms in the numerator on the same denominator and the whole thing basically solves itself! The only thing we need to realize is that tan(ℎ) ∕ ℎ = sin(ℎ) ∕ ℎ⋅1 ∕ cos(ℎ) which then clearly approaches 1 as ℎ approaches 0.
Great presentation! It was very clear and helpful. I also liked the last piece of advice. I agree 100% about the pursuit of learning. I strive to be a life-long learner. I am still studying at the age of 58 (taking a math class at my local university). My first four degrees were fun (BA,MA,MS,PhD) and my job is great, but learning new things every day is what really keeps me going. Thanks for this video, it was awesome to watch!
@@PrimeNewtonsThanks! I give the credit to family who instilled in me the value of education. Both of my grandfathers were from immigrant families arriving in the USA around 1905. My Swedish grandfather’s family immigrated to northern Minnesota and it was a hard life there. But my Minnesota grandfather worked at an ice cream factory in Minneapolis to help pay for his education at the University of Minnesota. He kept on getting degrees. He got a DVM and then a PhD in microbiology. As he was finishing his PhD, he was drafted to serve in WW II. Because he was older, and because of his education, the U.S. army decided to make him a captain even though he had no military experience. He was in Italy for years during WW II, and that was hard because he had to leave behind my grandmother and father. War is a nightmare, but he was lucky to get through it to return home. After the war he was offered many university positions but he decided on being a research scientist at the Mayo Clinic instead. Education brought my grandfather from being a poor immigrant who had to wear shoes that were falling apart to a life later on that he enjoyed. He was a great role model. The world needs great role models (teachers) to help the next generation live happy and well thought-out lives.
My mentor ❤🎉 Mr Prime ... you changed 60% of the students who had negative attitude toward maths...here in kenya 🇰🇪 😅😅😅 you will not remain the same Mr prime 🎉God blessed you🫡🫡🫡🫡
I used formula for tan(x+y) and had to taclulate limit limit(tan(h)/h,h=0) Finally I have got following answer (tan(x))' = 1+tan^2(x) and in my opinion this form of answer is more useful especially for integrals
It's 10 pm in Brazil and I just thought: "How is the derivative of Tan x by the first principle, and then I found this guy one of my favorites math YT channels" Thanks Prime... this video couldn't be clearer than it aleeady is, nice job
