To differentiate using the first principle, Here are the steps.

First, you make y increases by dy and x increase by dx

Secondly, you subtract y from both sides

Thirdly, you divide both sides by dx

Lastly, you apply the limit expression as dx tends to approach 0

The next three examples illustrate this;

Example 1

Given that y=x, Find $\frac{dy}{dx}$



Let y increase by $dy$ and x increase by $dx$.


Now, we subtract $-y$ from both sides


Since $y=x$



Now, let divide both sides $dx$



Example 2

From the first principle, differentiate  $y=x^2+4$


Let y increase by $dy$ and x increase by $dx$.




If $y=x^2+4$, then


Now, let distribute the negative sign




Applying limit expression

$$\frac{dy}{dx}=\lim_{dx \to 0} 2x+(0)$$


Example 3

Using first principle, find the $\frac{dy}{dx}$ of $y=x^2+x+4$



Let y increase by $dy$ and x increase by $dx$.


Expanding the expression


Now, let subtract y from both sides


If $y=x^2+x+4$, then


Now, let's distribute the negative sign



Dividing both sides by $dx$


Now, let's apply the limit expression

$$\frac{dy}{dx}=\lim_{dx \to 0}2x+0+1$$



That will be all for now. In the next post, we will be learning how to find the derivative of a function.

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