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…

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

Answer

Explanation:

Step1: Recall the formula for the derivative

The instantaneous rate of change of a function (y = f(x)) at a point (x=a) is given by (f^{\prime}(a)). For (y=\frac{u}{v}) (where (u = x^{7}-5x) and (v = 9)), by the quotient rule ((\frac{u}{v})^{\prime}=\frac{u^{\prime}v - uv^{\prime}}{v^{2}}). Since (v = 9), (v^{\prime}=0). So (f^{\prime}(x)=\frac{(7x^{6}-5)\times9-(x^{7}-5x)\times0}{9^{2}}=\frac{7x^{6}-5}{9}) (using the power rule ((x^{n})^{\prime}=nx^{n - 1}), so (u^{\prime}=(x^{7}-5x)^{\prime}=7x^{6}-5)).

Step2: Evaluate the derivative at (x = 1)

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

Answer:

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