Mathematical Background to Solve Physics Problems Derivatives

Derivatives

There are numerous definitions of the derivative, but the one that fits most physics problems best is that the derivative of a function is another function that gives the slope of the original function at any point. Consider a function f(x), often written as y = f(x), over an interval äx. The notation y = f(x) is mathematical symbolism that says “a variable y is going to be described by certain operations on another variable x.”

Using the ä notation the general expression for slope is

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This equation says that the slope of a function is the value of the function at a point x + ƒÂx minus the value of the function at x all divided by the ƒÂx. This assumes the function is linear between x and x + ƒÂx; an approximation that gets better as ƒÂx gets smaller. The slope defined this way is an average slope between x and x + ƒÂx. The derivative is the general expression for the slope at any point, thus, it is a function that gives the slope of another function at every point. The derivative, df/dx or f’ is the limiting case of the slope where ƒÂx ¨ 0

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Now apply this procedure to several functions.

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The function is a constant so f(x + δx) = f(x) and the slope is zero as is evident from the graph.

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The slope of the curve y = x2 – 5 is 2x. Just pick a value of x, and the slope is two times this value.

As an exercise verify that the derivative of y = x3 is 3×2.

The derivative of power law functions is very easy with the procedure described above. After performing a few of these, we can come to the conclusion that for any power law y = cvn, the general expression for the slope (derivative) is y’ = cnvn 1. Listed below are the derivatives for power laws as well as some trigonometric, exponential, and logarithmic functions. All of these can be derived using the procedures employed above.

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One other useful rule of differentiation is the chain rule. If y is written in terms of x and x is written terms of t, it is possible to write dy/dt through the simple expediency of a chain derivative.

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and since x is written in terms of t, the derivative dy/dt can be written in terms of x or t.

 

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