Question 1187895


Find the gradient of {{{f(x)}}} at {{{x=4}}} for which {{{f(x)=(5(x-1)(x-5))/(8x^(1/2))}}} or {{{f(x)=(5(x-1)(x-5))/(8sqrt(x))}}} 


The gradient (slope) of a curve of  {{{f(x)}}}  at {{{x=a }}}is {{{f}}}'{{{(a)}}}


By taking the derivative, {{{(d/dx)((5 (x - 1) (x - 5))/(8sqrt(x))) we get 


{{{f}}}'{{{(x) = (5 (3 x^2 - 6 x - 5))/(16 x^(3/2))}}}


By plugging in {{{x=4}}} we get


{{{f}}}'{{{(4) = (5 (3*4^2 - 6*4 - 5))/(16*4^(3/2))}}}

{{{f}}}'{{{(4) = 95/128}}}
 

Hence, the gradient is {{{95/128 }}} (exact solution) approximately {{{0.7421875}}}