# 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 $x$.