Determine equation of the tangent line to f (x) = 4x - 8 √x  at x = 16 .

Solution : We already know that the equation of a tangent line is specified by,

                           y = f ( a ) + f ′ ( a ) ( x - a )

Hence, we will have the derivative of the function (don't forget to get rid of the radical).

f ( x ) = 4x - 8x ^½    ⇒   f ′ ( x ) = 4 - 4x ^-1/2  = 4 -  4/x ^1/2

Again, notice as well that we remove the negative exponent in the derivative solely for the sake of the evaluation.  All we have to do then is evaluate the function & the derivative at the point in question, x = 16 .

f (16) = 64 - 8 ( 4) = 32                                                       f ′ ( x )  = 4 - 4 /4= 3

Then the tangent line is,

y = 32 + 3( x -16) = 3x -16