Using the limit definition, how do you differentiate #f(x) =4/(sqrt(x))#?
1 Answer
Jan 8, 2016
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
#=(-4)/(sqrtxsqrtx(sqrtx+sqrtx))#
#=(-4)/(x(2sqrtx)#
#=color(green)(-2/x^(3/2)#