simplify the difference quotients fxhfxh and fxfaxa for the

simplify the difference quotients (f(x+h)-f(x))/h and (f(x)-f(a))/x-a for the following function by rationalizing the numberator: f(x)=8 square rootX

Solution

f(x) = 8x

f(x+h) = 8(x +h)

(f(x+h) - f(x))/h = (8(x +h) - 8x)/h

= 8((x +h) - x)/h

multiply and divide by (x +h) + x

= 8((x +h) - x)/h * ((x +h) + x)/((x +h) + x)

= 8[(x+h) -x]/[h((x +h) + x)]           since (a- b)(a +b) = a² + b²

= 8/((x +h) + x)

f(a) = 8a

(f(x) - f(a))/(x -a) = (8x) - 8a)/(x -a)

= 8(x - a)/(x -a)

multiply and divide by x + a

= 8(x - a)/(x -a) * (x + a)/(x + a)

= 8(x -a)/[(x -a)(x + a)]

= 8/(x + a)