Examples of Derivative by Definition

Example:
Find, by definition, the derivative of function {x^2} - 1 with respect to x.

Solution:
Let

y = {x^2} - 1

I. Change x to x + \Delta x and y to y + \Delta y

y + \Delta y = {(x + \Delta x)^2} - 1

II. Find \Delta y by subtraction

\begin{gathered} \Delta y = {(x + \Delta x)^2} - 1 - y \\ \Delta y = {(x + \Delta x)^2} - 1 - ({x^2} - 1) \\ \Delta y = {(x + \Delta x)^2} - 1 - {x^2} + 1 \\ \Delta y = {x^2} + 2x\Delta x + {(\Delta x)^2} - {x^2} \\ \Delta y = 2x\Delta x + {(\Delta x)^2} \\ \end{gathered}

III. Divide both sides by \Delta x

\begin{gathered} \frac{{\Delta y}}{{\Delta x}} = \frac{{2x\Delta x + {{(\Delta x)}^2}}}{{\Delta x}} \\ \frac{{\Delta y}}{{\Delta x}} = \frac{{\Delta x(2x + \Delta x)}}{{\Delta x}} \\ \frac{{\Delta y}}{{\Delta x}} = (2x + \Delta x) \\ \end{gathered}

IV. Find the limit of \frac{{\Delta y}}{{\Delta x}} where \Delta x \to 0

\begin{gathered} \mathop {\lim }\limits_{\Delta x \to 0} \frac{{\Delta y}}{{\Delta x}}\mathop { = \lim }\limits_{\Delta x \to 0} (2x + \Delta x) \\ \Rightarrow \frac{{dy}}{{dx}} = (2x + 0) \\ \Rightarrow \frac{{dy}}{{dx}} = 2x \\ \end{gathered}

This is the derivative of y = {x^2} - 1 w.r.t x.

Example:

Find, by definition, the derivative of function \frac{1}{{x + a}} with respect to x.

Solution:
Let

y = \frac{1}{{x + a}}

I. Change x to x + \Delta x and y to y + \Delta y

y + \Delta y = \frac{1}{{x + \Delta x + a}}

II. Find \Delta y by subtraction

\begin{gathered} \Delta y = \frac{1}{{x + \Delta x + a}} - y \\ \Delta y = \frac{1}{{x + \Delta x + a}} - \frac{1}{{x + a}} \\ \Delta y = \frac{{x + a - (x + \Delta x + a)}}{{(x + \Delta x + a)(x + a)}} \\ \Delta y = \frac{{x + a - x - \Delta x - a}}{{(x + \Delta x + a)(x + a)}} \\ \Delta y = \frac{{ - \Delta x}}{{(x + \Delta x + a)(x + a)}} \\ \end{gathered}

III. Divide both sides by \Delta x

\begin{gathered} \frac{{\Delta y}}{{\Delta x}} = \frac{{ - \Delta x}}{{\Delta x(x + \Delta x + a)(x + a)}} \\ \frac{{\Delta y}}{{\Delta x}} = \frac{{ - 1}}{{(x + \Delta x + a)(x + a)}} \\ \end{gathered}

IV. Find the limit of \frac{{\Delta y}}{{\Delta x}} where \Delta x \to 0

\begin{gathered} \mathop {\lim }\limits_{\Delta x \to 0} \frac{{\Delta y}}{{\Delta x}} = \mathop {\lim }\limits_{\Delta x \to 0} \frac{{ - 1}}{{(x + \Delta x + a)(x + a)}} \\ \Rightarrow \frac{{dy}}{{dx}} = \frac{{ - 1}}{{(x + 0 + a)(x + a)}} \\ \Rightarrow \frac{{dy}}{{dx}} = \frac{{ - 1}}{{(x + a)(x + a)}} \\ \Rightarrow \frac{{dy}}{{dx}} = \frac{{ - 1}}{{{{(x + a)}^2}}} \\ \end{gathered}

This is the derivative of y = \frac{1}{{x + a}} w.r.t x.