Question

Using the limit definition, how do you differentiate `f(x)=1-x^2`?

Answer 1

If `f(x) = 1-x^2` then `(d f(x))/(dx) = -2x`

Explanation:

The limit definition of the derivative of a function `f(x)` is
`color(white)("XXX")(d f(x))/(dx) = lim_(hrarr0) (f(x+h)-f(x))/h`

If `f(x) = 1-x^2`
`color(white)("XXX")`then (by the limit definition)
`(d f(x))/(dx)`
`color(white)("XXX")=lim_(hrarr0) ((1-(x+h)^2)-(1-x^2))/h`

`color(white)("XXX")= lim_(hrarr0) (1-x^2-2xh-h^2-1+x^2)/h`

`color(white)("XXX")=lim_(hrarr0)(-2xh-h^2)/h`

`color(white)("XXX")=lim_(hrarr0) -2x-h`

`color(white)("XXX")=-2x`