Question

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

Answer 1

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`