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. 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 the $f'(x)$ .

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.$ (Error compiling LaTeX. Unknown error_msg)g'(x)=t \cdot x^{t-1}+(n-1) \cdot x^{n-2}+ 5 \cdot 3 \cdot x^2$$ (Error compiling LaTeX. Unknown error_msg)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$ (Error compiling LaTeX. Unknown error_msg)f(x) $is a constant then its derivative will always be$ 0$.

Notation:$ (Error compiling LaTeX. Unknown error_msg)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$ (Error compiling LaTeX. Unknown error_msg)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.

Below are problems for Part I. In Part II(see link below) we will begin to actually "start" the calculus with this.

== Problems for Part I ==$ (Error compiling LaTeX. Unknown error_msg)\boxed{\text{Problem 1}} $: Find the first derivative of$ f(x) $, where$ f(x)=2x^2-15x+7 $.$ \boxed{\text{Solution 1}} $:$ f'(x)=2 \cdot 2 \cdot x^1-15 \cdot 1 \cdot x^0+0$$ (Error compiling LaTeX. Unknown error_msg)f'(x)=4x-15 $.$ \boxed{\text{Problem 2}} $: Find the equation of the line tangent to the function$ f(x)=3x^3-5x^2+12 $at$ (-1,14) $.$ \boxed{\text{Solution 2}}$:

We will take the first derivative to determine the slope of the tangent line.$ (Error compiling LaTeX. Unknown error_msg)f'(x)=9x^2-10x $. If this is the slope of the tangent point then we can just plug$ -1 $into the$ x $coordinate to find the actual slope.$ f'(x)=9+10=19 $. The slope of the line is$ 19$.

Let the equation be:$ (Error compiling LaTeX. Unknown error_msg)y=19x+b$.

Plugging$ (Error compiling LaTeX. Unknown error_msg)(-1,14) $in gives:$ 14=-19+b$$ (Error compiling LaTeX. Unknown error_msg)\implies b=30 $.$ \therefore $The equation of the line is$ y=19x+33 $.$ \boxed{\text{Problem 3}} $: Find the nth derivative of$ f(x)=x^n$$ (Error compiling LaTeX. Unknown error_msg)\boxed{\text{Solution 3}} $:$ \frac{d}{dx} f(x)=nx^{n-1}$$ (Error compiling LaTeX. Unknown error_msg)\frac{d^2}{dx^2} f(x)=n(n-1) x^{n-2}$$ (Error compiling LaTeX. Unknown error_msg)\vdots$$ (Error compiling LaTeX. Unknown error_msg)\frac{d^{n}}{dx^{n}} f(x)=n(n-1)(n-2) \cdots 1$$ (Error compiling LaTeX. Unknown error_msg)\frac{d^{n}}{dx^{n}} f(x)=n!$$ (Error compiling LaTeX. Unknown error_msg)\therefore $The nth derivate of$ f(x) $is$ n!$.

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Derivative/Definition