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Answer 1 · 2017-01-27T02:41:33

There is more than one method called the three-step method. (The steps are described differently.) They have the following in common.

#f(x) = 2x^3#

We'll need to find #f(x+h)# and #f(x+h)-f(x)# and #(f(x+h)-f(x))/h#

Some call this steps 1 and 2, others call all of this step 1.

#f(x+h) = 2(x+h)^3 = 2(x^3+3x^2h+3xh^2+h^3)#

# = 2x^3+6x^2h+6xh^2+2h^3#

#f(x+h)-f(x) = 2(x+h)^3 - 2x^3#

# = 2x^3+6x^2h+6xh^2+2h^3-2x^3#

# = 6x^2h+6xh^2+2h^3#

#(f(x+h)-f(x))/h = (6x^2h+6xh^2+2h^3)/h#

# = (h(6x^2+6xh+2h^2))/h#

# = 6x^2+6xh+2h^2#

Now we need to find #lim_(hrarr0)(f(x+h)-f(x))/h#

#lim_(hrarr0)(6x^2+6xh+2h^2) = 6x^2+0+0 = 6x^2#

So #f'(x) = 6x^2#