How do you use the formal definition to find the derivative of #y=1-x^3# at x=2?

1 Answer
Aug 21, 2015

That depends on which formal definition of the derivative at #x=a# you are using.

Explanation:

Using definition #lim_(hrarr0)(f(2+h)-f(2))/h#

#lim_(hrarr0)(f(2+h)-f(2))/h = lim_(hrarr0)([1-(2+h)^3]-[1-(2)^3])/h#

# = lim_(hrarr0)([1-(8+12h+6h^2+h^3)]-[1-8])/h# #" "#See Note below

# = lim_(hrarr0)(-12h-6h^2-h^3)/h#

# = lim_(hrarr0)(h(-12-6h-h^2))/h#

# = lim_(hrarr0)(-12-6h-h^2)#

# = -12#

Note: expand #(2+h)^3# using the binomial expansion or by multiplying #(2+h)(2+h)(2+h)#

Using definition #lim_(xrarr2)(f(x)-f(2))/(x-2)#

#lim_(xrarr2)(f(x)-f(2))/(x-2) = lim_(xrarr2)([1-x^3]-[1-2^3])/(x-2)#

# = lim_(xrarr2)(-x^3+8)/(x-2)#

# = lim_(xrarr2)(-(x^3-8))/(x-2)#

# = lim_(xrarr2)(-(x-2)(x^2+2x+4))/(x-2)#

# = lim_(xrarr2)-(x^2+2x+4)#

# = -12#