How to find the derivative of an equation

Calculus Tutorial

How to find the derivative of an equation

Intro

In Calculus, you will need to know how to find the derivative of an equation to solve many problems. So here is how to find the derivative effectively!

Sample Problem

f(x) = 4x^2 + 8x
f'(x) = ?

16x+8


8x


16x


8x+8


12x^2


Solution

Let’s start by talking about the equation.
f(x) (pronounced “f of x”) is just fancy notation for “y”. It’s really the value of y when x=x. So for example, on this graph below, f(2) = 3 or when x = 3, y = 2.
Simple linear graph
So what does the graph of f(x) = 4x^2 + 8x + 6 look like? Well, a bit like this:
parabola
The derivative of this graph would be a line tangent to this parabola. Now, I could go through the original derivative formula, but its long, confusing, and doesn’t do much to help you understand derivatives so I’ll be showing you the easy way to do it.

First use the notation f'(x) (pronounced “f prime of x”) to indicate that your new equation is a derivative of f(x).

1. When faced with a variable with an exponent (ex: 4x^2), multiply the coefficient (the number in front of the variable) with the exponent, then subtract 1 from the exponent. You will end up with 8x^1 or just 8x.

2. The derivative of any number with a variable raised to the first power will just be the coefficient. (ex: f(x) = 8x, f'(x) = 8)

3. The derivative of any standalone number is always zero. Derivative of 6? Zero. Derivative of 6,000,000? Zero. If it has no variable, it’s zero.

So, f'(x) = 8x + 8. Easy enough? Great, let’s get into some rules of derivatives!

The first thing you should know about derivatives is that you can derive as many times as you like. You can find the derivative of f'(x) and call it f”(x)! For our sample equation, f”(x) = 8. And you could just keep going. Not really that far with this equation, but with others, you definitely can.
For example:
f'(x) = 3x^4 + 2x^2 + 7x + 5
f”(x) = 12x^3 + 4x + 7
f”'(x) = 36x^2 + 4
f^(4)(x) = 72x
f^(5)(x) = 72
f^(6)(x) = 0
It’s a lot, I know. Hopefully you never have to go that far.



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