The generalized product rule ( Leibniz Formula )

The generalized product rule at the face of it not intuitive and that’s probably why a lot of people make mistakes when asked to differentiate functions n-times.

If f and g are functions that are continuous and differentiable everywhere, then:

Now, we would like to find out a generalized expression for the n-th derivative of fg.

In order to arrive at that formulation lets calculate a few derivatives to see whether we can find a pattern:





You must have noticed a pattern in the above expressions. The coefficients seem are the one in the binomial expansion of (x+y)n

Therefore we can write the expression for the n-derivative of fg as the following:

where (i) means to differentiate i-times. This is also known as Leibniz Formula.

Have a good one!

High Lady of the Night Court

So I don’t usually post what I draw cuz can’t really draw. But I kind of liked how this one came out, so I am throwing it out into the black void of tumblr. 

Heavily inspired by that one drawing you did of a smiling Phillip with a flower crown, this is Moana! (still trynna get the hang of the whole digital art thing its so tough. Any tips???) 💕💕

Everyone always makes jokes about libras being unable to answer simple questions without knowing how the other person would answer first (that post where someone’s like “hey how are you” libra goes idk how are you? Person says fine and libra says Oh I’m fine too). Anyways I never really fully understood it until the other day I asked my libra dom (Mercury Venus 5h) friend if he liked crunchy or smooth peanut butter better, and I’ve never seen someone struggle so much with that question in my entire life LMAO he tried to get me to say what MY favorite was and I was like lmao no way I asked first! It literally took him at least 5 minutes to answer fjfjfjgiiffijffhfbb