# Examples of Derivative by Definition

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

Solution:
Let

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

II. Find $\Delta y$ by subtraction

III. Divide both sides by $\Delta x$

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

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

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

II. Find $\Delta y$ by subtraction

III. Divide both sides by $\Delta x$

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

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