Engineering Calculus Notes 255

Engineering Calculus Notes 255 - 3.3. DERIVATIVES 243...

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Unformatted text preview: 3.3. DERIVATIVES 243 Similarly, if g is a differentiable real-valued function of a real variable and f is a differentiable vector-valued function of a real variable, the composition f g is another vector-valued function, whose derivative is the product of the derivative of f ( i.e. , , its velocity) and the derivative of g : d dt vextendsingle vextendsingle vextendsingle vextendsingle t = t bracketleftBig f g bracketrightBig = vector f ( g ( t )) g ( t ) . We would now like to turn to the case when g is a vector-valued function of a real variable, and f is a real-valued function of a vector variable, so that their composition f g is a real-valued function of a real variable. We have already seen that if g is steady motion along a straight line, then the derivative of the composition is the same as the action of the differential of f on the derivative ( i.e. , the velocity) of g . We would like to say that this is true in general. For ease of formulating our result, weto say that this is true in general....
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