1 = 2Īnd indeed, the difference between 10^2 and (10.1)^2 is about 2. The output moves 20 units for every unit of input movement. At x = 10 the "output wiggle per input wiggle" is = 2 * 10 = 20.If our input lever is at x = 10 and we wiggle it slightly (moving it by dx=0.1 to 10.1), the output should change by dy. This is usually a formula, not a static value, because it can depend on your current input setting.įor example, when f(x) = x^2, the derivative is 2x. The result can be written "output wiggle per input wiggle" or "dy/dx" (5mm / 1mm = 5, in our case).
"Oh, we moved the input lever 1mm, and the output moved 5mm. The lever is at x, we "wiggle" it, and see how y changes. See any graphs around these parts, fella?) What does that mean? (And don't mindlessly mumble "The derivative is the slope". The derivative is the "moment-by-moment" behavior of the function. Another analogy: x is the input signal, f receives it, does some magic, and spits out signal y. Think of function f as a machine with an input lever "x" and an output lever "y". The machine computes functions like addition and multiplication with gears - you can see the mechanics unfolding! Check out this incredible, mechanical targetting computer ( beginning of youtube series). I visualize a function as the process "input(x) => f => output(y)".
But the derivative rules are about the machinery, so let's see it! Graphs squash input and output into a single curve, and hide the machinery that turns one into the other. The default calculus explanation writes "f(x) = x^2" and shoves a graph in your face. Onward! Functions: Anything, Anything But Graphs This installment covers addition, multiplication, powers and the chain rule. The goal is to really grok the notion of "combining perspectives". Instead of memorizing separate rules, let's see how they fit together: Each derivative rule is an example of merging various points of view.Īnd why don't we analyze the entire system at once? For the same reason you don't eat a hamburger in one bite: small parts are easier to wrap your head around. Combine every point of view to get the overall behavior. Every part has a "point of view" about how much change it added. Using the behavior of the parts, can we figure out the behavior of the whole?.It turns out f is part of a bigger system (h = f + g).The derivative f' (aka df/dx) is the moment-by-moment behavior.
We have a system to analyze, our function f.
The addition rule, product rule, quotient rule - how do they fit together? What are we even trying to do? In this way, we only look at the partial variation in the function instead of the total variation.įor example, if a function is \(f(x,y)\), we use the partial differentiation with respect to \(x\), to measure the rate of change in \(f(x,y)\) when only \(x\) changes and \(y\) remains constant.The jumble of rules for taking derivatives never truly clicked for me. To apply the rules of calculus, at a time generally, we change only one independent variable and keep all other independent variables constant. If a function is a multivariable function, we use the concept of partial differentiation to measure the effect of a change in one independent variable on the dependent variable, keeping the other independent variables constant. Section 2 Partial derivatives and the rules of differentiation
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