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

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

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

Answer

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})). [ \begin{align*} f^{\prime}(x)&=\frac{1}{9}(x^{7}-5x)^\prime\ &=\frac{1}{9}(7x^{6}-5) \end{align*} ]

Step3: Evaluate the derivative at (x = 1)

Substitute (x = 1) into (f^{\prime}(x)): [ \begin{align*} f^{\prime}(1)&=\frac{1}{9}(7\times1^{6}-5)\ &=\frac{1}{9}(7 - 5)\ &=\frac{2}{9} \end{align*} ]

Answer:

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