Using the limit definition, how do you differentiate #f(x) =4/(sqrt(x))#?

1 Answer
mason m
Jan 8, 2016

#f'(x)=-2/x^(3/2)#

Explanation:

The limit definition of the derivative:

#lim_(hrarr0)(f(x+h)-f(x))/h#

Thus, the derivative can be found through

#f'(x)=lim_(hrarr0)(4/sqrt(x+h)-4/sqrtx)/h#

#=lim_(hrarr0)(4/sqrt(x+h)-4/sqrtx)/h*(sqrtx * sqrt(x+h))/(sqrtx * sqrt(x+h))#

#=lim_(hrarr0)(4sqrtx-4sqrt(x+h))/(hsqrtxsqrt(x+h))#

#=lim_(hrarr0)(4(sqrtx-sqrt(x+h)))/(hsqrtxsqrt(x+h))*(sqrtx+sqrt(x+h))/(sqrtx+sqrt(x+h))#

#=lim_(hrarr0)(4(x-(x+h)))/(hsqrtxsqrt(x+h)(sqrtx+sqrt(x+h)))#

#=lim_(hrarr0)(-4h)/(hsqrtxsqrt(x+h)(sqrtx+sqrt(x+h)))#

#=lim_(hrarr0)(-4)/(sqrtxsqrt(x+h)(sqrtx+sqrt(x+h)))#

Now, plug in #0# for #h#.

#=(-4)/(sqrtxsqrtx(sqrtx+sqrtx))#

#=(-4)/(x(2sqrtx)#

#=color(green)(-2/x^(3/2)#

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