Derivative/Definition

Differential Calculus is a sub-field of Calculus that primarily focuses on how functions change as the input changes. In Differential Calculus we usually use Differentiation, or the process of finding the derivative.

Derivative represents the slope of the slope of the line tangent to a function at some point.

Long method for Derivative: Let the function be $f(x)=ax^n+bx^{n-1}+cx^{n-2}+ \cdots z=0$ . Find the First Derivative.

$\boxed{\text{Solution:}}$ If we imagine the secant line intersecting a curve at the points $A$ and $B$ . Then we can change this to the tangent by setting $B$ on top of $A$ . Let us call the horizontal or vertical distance as $h$ .

$\lim_{h\to0} \frac{f(x+h)-f(x)}{h}$

$\implies \lim_{h\to0} \frac{a(x+h)^n+b(x+h)^{n-1}+c(x+h)^{n-2}+ \cdots z-(ax^n+bx^{n-1}+cx^{n-2}+ \cdots z)}{h}$

After canceling like terms we should have all terms contain an $h$ . We can then cancel out the $h$ and set $h=0$ . Our end result is the first-derivative.

The first derivative is denoted as $f'(x)$ .

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Derivative/Definition