Question

1. if ( f ) is the function defined by ( f(x)=\frac{x^{7}-5 x}{9} ), then the instantaneous rate of change of ( f ) at ( x = 1 ) is

(a) ( \frac{2}{9} )
(b) ( \frac{4}{9} )
(c) ( \frac{7}{9} )
(d) ( \frac{10}{9} )

Explanation:

Step1: Recall the formula for instantaneous rate of change

The instantaneous rate of change of a function \(y = f(x)\) at a point \(x=a\) is given by \(f^{\prime}(a)\), where \(f^{\prime}(x)\) is the derivative of \(f(x)\).

Step2: Differentiate the function \(f(x)=\frac{x^{7}-5x}{9}\)

Using the sum - difference rule \((u\pm v)^\prime=u^\prime\pm v^\prime\) and the power rule \((x^{n})^\prime = nx^{n - 1}\), and the constant multiple rule \((cf(x))^\prime=c f^{\prime}(x)\) (where \(c=\frac{1}{9}\)).

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Step3: Evaluate the derivative at \(x = 1\)

Substitute \(x = 1\) into \(f^{\prime}(x)\):

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Answer:

A. \(\frac{2}{9}\)