Differentiate the following

Question

Use the limit definition of the derivatives to differentiate the following: CHeck answer using the power rule.

a. f(x)= x^2 + 9x - 2

b. g(x) = 8 / square root of x

Answer 1

Let L = lim_h→0

a)

f'(x) = L((x+h)²+9(x+h)-2 - x²-9x+2)/h

= L(x²+2hx+h²+9x+9h-2-x²-9x+2)/h

= L(2hx+h²+9h)/h

= L(2x+h+9)

= 2x+9

b)

g'(x) = L(8/√(x+h) - 8/√x)/h)

= L(((8√x-8√(x+h))/(√x√(x+h))/h)

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

= L((64x-64x-64h)/((h√x√(x+h)(8√x+8√(x+h))

= L(-64h/(h√x√(x+h)(8√x+8√(x+h))

= L(-64/(√x√(x+h)(8√x+8√(x+h))

= -64/(√x√x(8√x+8√x)

=-64/(16√x√x√x)

= -4/x^3/2