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Introduction to Derivatives

What are derivatives?

In Math, derivatives represent the function of the slope of a function. We all know that rise over run (y1-y2)/(x1-x2) gives us the slope of a function - but what if we need to find out the slope of a function at a specific point? This is when derivatives come into play.


As an example, for the function f(x) = x^2, the derivative is f’(x) = 2x. What does this specifically mean? When asked what the slope for f(x) is at a given point, we can easily get the answer by plugging the number into f’(x) = 2x. That being said, the slope of f(x) at x = 5 is 10, as 2*5 = 10.



Basic derivative rules

Now, how do we take derivatives? Below is the introduction to some basic derivative rules.


The constant rule

The constant rule states that the derivative of any constant function is zero. For example, if f(x) = 5, f’(x) = 0.


The power rule

The derivative of a function raised by power “a” is “a” times the function raised to the power of “a-1”. For instance, if f(x) = x^3, then f’(x) = 3x^2; if f(x) = x^6, then f’(x) = 6x^5.


Because of the power rule, we can conclude that the derivative of f(x) = x is 1. This is because x has a power of 1, so the derivative is 1*x^0. Because anything with a power of 0 is 1, the derivative of f(x) = x is 1*1, which is just 1.


The constant multiple rule

If a constant is multiplied by a function, the derivative is the constant multiplied by the derivative of the function. For instance, if f(x) = 4x, f’(x) = 4*1 = 4.


The sum and difference rules

If two or more functions are added to or subtracted from one another, the derivative is simply the sum or difference of the derivative of each function. For instance, if f(x) = 2x^2 + 4x, in this case, there are two functions adding together: 2x^2 and 4x. To find the derivative of f(x), we take the derivative of each individual function first. In this case, 2x^2 becomes 4x (power rule), and 4x becomes 4 (constant multiple rule). Therefore, f’(x) is the two derived functions added together, which is 4x + 4.


The aforementioned five rules are just some basic rules of taking derivatives. There are many other rules that you can continuously learn and explore!


Hope this article helped you learn more about derivatives. Thank you so much for reading!


 

Sources:


Introduction to derivatives. Math is Fun. (n.d.). Retrieved January 29, 2022, from https://www.mathsisfun.com/calculus/derivatives-introduction.html


Khan Academy. (n.d.). Basic differentiation review (article). Khan Academy. Retrieved January 29, 2022, from https://www.khanacademy.org/math/old-ap-calculus-ab/ab-derivative-rules/ab-basic-diff-rules/a/basic-differentiation-review



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