ABSTRACT
In elementary calculus, the main mathematical tool for studying the behavior of functions is the derivative. This state of affairs persists when we examine functions from one vector space to another. When we work in this more general setting, however, we need to change our notion of the derivative. If f : R → R https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315138695/b832c43e-68c9-4052-aad2-0d41b672a31d/content/eq912.tif"/> is a function, the derivative of f at a point x 0 is a number f ′ ( x 0 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315138695/b832c43e-68c9-4052-aad2-0d41b672a31d/content/eq913.tif"/> giving the slope of the graph of f at x 0. However, we can also view f ′ ( x 0 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315138695/b832c43e-68c9-4052-aad2-0d41b672a31d/content/eq914.tif"/> as defining a linear transformation L : R → R https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315138695/b832c43e-68c9-4052-aad2-0d41b672a31d/content/eq915.tif"/> , given by L ( x ) = f ′ ( x 0 ) x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315138695/b832c43e-68c9-4052-aad2-0d41b672a31d/content/eq916.tif"/>
