Apply the nice Chain Rule: 
to 
where:
and 
then
and 
so by the Chain Rule:
Archive for November, 2008
The Chain Rule
Sunday, November 23rd, 2008Differentiating from the start
Sunday, November 2nd, 2008Differentiation is, for me, where understanding of what’s going on slips into a mess of tools, symbols and formulas. The grapher application helps greatly with this – after doing the necessary mental gymnastics, it’s useful to see the relationship between the derivation and the original. Here’s 
and its derivative 
. You can see the latter changing nicely with the former.
The material I’m revising introduces differentiation from its bare definition, calculating the gradient between two different points on a smooth function, then making those points arbitrarily close from the quotient:
Then there’s lots of nice reasoning to find results for different functions, with new techniques enabling further discoveries. For example, here’s the logic for the sine function:
If 
and 
is more than zero, and 
, after collecting like terms we end up with:
it can be shown (using geometric and algebraic arguments) that:
and 
which ends up with:
leaving 
Cildo Meireles
Sunday, November 2nd, 2008Thursday 31st September – Cildo Meireles at the Tate Modern. Brazilian conceptual artist.
Information
Meireles spread ideas, statements and the art itself through use of existing mechanisms, such as banknotes, recycled coke bottles and the classified pages of newspapers. The choice of the system itself carries meaning (”Yankees go home!” on the bottles). The world around becomes the space for the art to work in.Babel (2001) is a tower of radios. Each one is a different model, tuned to different frequencies. Instead of being an overwhelming wall of noise, the many fuse into a single experience – you have to lean close to hear individual voices. It exposes a broad range of the electromagnetic radio spectrum, to be heard as a single thing.In Eureka/Blindhotland (1970-5) there are many spheres, which you are encouraged to pick up. Each is a different weight, from 500 grams to 1500 grams, but all are visually identical.Glovetrotter (1991) is also composed of spheres, but this time each is a different type. A steel mesh has been draped over all, concealing their identity, but not the scale.
Measuring space
The first room is full of precise works on graph paper, where repeated, systematically joined points create clearly defined volumes. This moves on to proposals/instructions for conceptual works, also drawn on graph paper, many of which concern different forms of measurement. For example, a work is proposed where Meireles will canoe around the North Pole, becoming a day younger on each circumnavigation. Maps feature heavily, fundamentally being the lines which delineate borders between countries.Fontes (1992/2008) is a brightly lit room full of tape measures hung from the ceiling, creating a forest in which visitors can lose themselves. There are spiral paths through the measures, which can be walked through – but becoming progressively narrower towards the middle of the room, and then imperceptible. Having followed the spiral, you are disorientated, and wanting to leave, it is very difficult to tell if you are pushing through the tapes in a straight line. When out, you are confronted with walls packed with clocks – each of which has a non-uniform distribution of time, with the 12,3,6 and 9 placed at varying points around the dial.