Question

Let `f(x) = -3 x^3 + 9 x + 4`, how do you use the limit definition of the derivative to calculate the derivative of f?

Answer 1

The definition is:

`f'(x)=lim_(hrarr0)(f(x+h)-f(x))/h`.

So:

`lim_(hrarr0)(-3(x+h)^3+9(x+h)+4-(-3x^3+9x+4))/h=`

`=lim_(hrarr0)(-3(x^3+3x^2h+3xh^2+h^3)+9x+9h+4+3x^3-9x-4)/h=`

`=lim_(hrarr0)(-3x^3-9x^2h-9xh^2-3h^3+9x+9h+4+3x^3-9x-4)/h=`

`=lim_(hrarr0)(-9x^2h-9xh^2-3h^3+9h)/h=`

`=lim_(hrarr0)(-9x^2-9xh-3h^2+9)=-9x^2+9`,

that is the derivative of the function.