Derivative/Definition

Differential Calculus Part I

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. We can also find critical points with the first and second derivative.

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 $f'(x)$ .

This would be some tedious work so instead there is a much nicer way to find the derivative.

Let $f(x)=3x^n$ . Let $g(x)=x^t+x^{n-1}+5x^{3}$

1. Find $f'(x)$ .

Any function like this is: $f'(x)=3 \cdot n \cdot x^{n-1}=3n \cdot x^{n-1}$

2. Find $g'(x)$ .

Breaking apart on what we used above.

$g'(x)=t \cdot x^{t-1}+(n-1) \cdot x^{n-2}+ 5 \cdot 3 \cdot x^2$

$g'(x)=t \cdot x^{t-1}+(n-1) \cdot x^{n-2}+15x^2$

Let $f(x)=-147$ . Find $f'(x)$ .

If the function $f(x)$ is a constant then its derivative will always be $0$ .

Notation: $f'(x)$ denotes the first derivative for $f(x)$ . The symbol for the second derivative is just $f''(x)$ . For the third derivative it is just $f'''(x)$ . Derivatives are also written as $\frac{d}{dx} f(x)$ . Or if for the nth derivative they are written as $\frac{d^n}{dx^n} f(x)$ .

Maximum and Minimum: We can use the first derivative to determine the maximum and the minimum points of a graph.

If $f'(x)=6x^2-24$ . Then the maximum and the minimum occur when:

$6x^2-24=$ , $x=2$ or $x=-2$ . We can plug each back in to the original $f(x)$ if it was given, and the one with the higher y-coordinate is the maximum, while the smaller y-coordinate gives the minimum.