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Unformatted text preview: Notes on the Development of the Derivative Transformation in Math 51 1. Single Variable Derivatives We are accustomed to saying that a single variable function f is “differentiable at the point a ” if the limit: lim h → f ( a + h )- f ( a ) h exists. The motivation behind this is that the existence of this limit is what ensures that there exists a tangent line to the graph of f at the point a ; in other words, a line that passes through the graph at the point a and which furthermore is a “good” approximation to the function near the point a . By a “good” approximation, we mean that the relative error at a point x approaches zero as x approaches a . To be precise: lim x → a | (error) | | Δ x | = 0 If we suppose that the slope of the tangent line in question is m and thus that the equation of that line is y = f ( a ) + m ( x- a ), then the limit above can be rewritten as lim x → a | ( f ( x ))- ( f ( a ) + m ( x- a )) | | x- a | = lim x → a | ( f ( x ))- f ( a )- m ( x- a ) | | x- a | = 0 Some algebraic manipulations show that this tangent line exists and satisfies the above condition if and only if the expression m = lim h → f ( a + h )- f ( a ) h exists. So, interestingly, the expression above actually serves two purposes for single variable functions – first of all, its existence determines whether or not there is a “good” linear approximation to the function near the point a (in other words, whether or not f is dif- ferentiable); second, if it does exist, then its value is equal to the slope of that line. We call this expression the “derivative of f at a ”, which we write as df dx ( a ) = lim h → f ( a + h )- f ( a ) h This derivative can be interpreted in other ways. For one thing, the fact that it is the slope of the line approximating f tells us that for small values of Δ x , we have Δ f ≈ df dx ¶ Δ x 1 Similarly, the chain rule tells us that if we view the value of x as changing over time, then df dt = df dx ¶ dx dt So the derivative can be viewed as (a) the thing you multiply by the change in x to get the change in f ; or (b) the thing you multiply by the speed of x to get the speed of f 2. Partial Derivatives Approaching multivariable functions for the first time, our first instinct is to try to find an expression like the single variable derivative above, since that proved to be so useful in that context. The most obvious way to do this is to simply view the multivariable function as a single variable function by deciding to view all variables as constants, except for one. The result is a single variable function, and we can write down its derivative exactly as we did before. If we choose the i th variable to be the one variable, we get lim h → f (-→ a + h-→ e i )- f (-→ a ) h Since this expression is precisely the derivative of a single variable function, we imme- diately conclude the same interpretations that we had for single variable derivatives....
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