Definition of derivative

Definition of derivative

We have studied the notion of average rate of change thus far, for example, change in position over time (velocity), average change in velocity over time (acceleration) etc. However, what if we are interested in finding the instantaneous rate of change of something? To answer this, we will first learn about the concept of the definition of derivative in this section, as well as how to apply it.
Basic Concepts: Composite functions


Definition of Derivative
f(x)=  h0  limf(x+h)f(x)hf'\left( x \right) = \;_{h \to 0}^{\;lim}\frac{{f\left( {x + h} \right) - f\left( x \right)}}{h}
  • 1.
    Find the derivative of the given function using the definition of derivative.
    f(x)=x35x+6f\left( x \right) = {x^3} - 5x + 6

  • 2.
    Definition of derivative with irregular functions
    Find the derivative of the following functions using the definition of derivative.
    f(x)=x2 f(x)=\sqrt{x-2}

    f(x)=3x2+x f(x)=\frac{3-x}{2+x}

  • 3.
    Applications to definition of derivative
    Let f(x)=4x13f(x)=4x^{\frac{1}{3}}
    For when x x \, 0 \, 0 , find the derivative of f(x)f(x).

    Show that f(0)f'(0) does not exist.

    For what value(s) of xx does the vertical tangent line occur?