For , let be a positive real number. Answer the following in order to obtain an…

MATHEMATICS · Multi-part · CBSE · Mathematics · Limits and Derivatives · Limits and Derivatives

Problem

For

f(x)=x^{7/3}

, let

x

be a positive real number. Answer the following in order to obtain an explicit formula for

f'(x)

and to analyze differentiability of the function. (a) Write an algebraic expression for

\dfrac{f(x+h)-f(x)}{h}

and simplify the numerator as much as possible without taking the limit. (b) Determine

\displaystyle \lim_{h\to0} \dfrac{f(x+h)-f(x)}{h}

and hence obtain a formula for

f'(x)

. (c) Identify every interval of

x

on which the derivative you found in part (b) exists, and justify your conclusion.