Derivatives

# Derivatives - Fig 1 The derivative of a function is the...

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Fig. 1: The derivative of a function is the slope of its graph. Derivatives From Physics Notes Here is a brief summary of the differential calculus. Contents 1 Derivatives 2 Taylor series 2.1 Useful Taylor series 3 Ordinary differential equations 4 Partial derivatives 4.1 Change of variables 4.2 Partial differential equations 5 Exercises Derivatives The derivative of a function is another function, defined in terms of a "limiting expression": If the limiting expression give a unique and well-defined result within some domain of , then we say that the derivative "exists" in that domain. We also say that is differentiable in that domain. (It can be shown that a differentiable function is automatically continuous. Try proving it!) For the purposes of dimensional analysis, the derivative of a function has the units of the original function, divided by the units of . (This is obvious from the definition.) Second-order and higher-order derivatives are defined by repeating the derivative procedure. Graphically, the derivative represents the "slope" of the graph of , while the second derivative represents the "curvature". For example, the graph in Fig. 1 has positive second derivative, because it is upward-curving. Derivatives obey several elementary composition rules: These can all be proven by direct substitution into the definition of the derivative, and taking appropriate orders of limits. With the aid of these rules, we can prove various standard results, such as the "power rule" for derivatives: . The linearity of the derivative operation has the important implication that derivatives "commute" with sums, i.e. you can move them to the left or right of summation signs. For example, we can use this to show that the exponential function is its own derivative:

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Derivatives also "commute" with limit expressions. For example, we can use this on the alternative definition of the exponential function: Taylor series A function is "infinitely differentiable" if all orders of derivatives are well-defined (i.e., first derivative, second derivative, etc.). Not all functions behave this way: for example, has a first derivative which is discontinuous at , which means that it has no well-defined second derivative at that point. If a function is infinitely differentiable, then near any point it can be written out in a "Taylor series": Here, the "zeroth derivative" refers to the function itself. The Taylor series can be derived by assuming that can be written out
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