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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