How to Calculate a Basic Derivative of a Function

This is intended as a guide to assist those who must occasionally calculate derivatives in generally non-mathematical courses such as economics, and can also be used as a guide for those just starting to learn calculus. This guide is meant for those who are already comfortable with algebra.
Note: The symbol for a derivative used in this guide is the ' symbol, * is used for multiplication, and ^ indicates an exponent.

Part 1

Part 1 of 2:

Starting with the Basics

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For instance, if you have a function that describes how fast a car is going from point A to point B, its derivative will tell you the car's acceleration from point A to point B—how fast or slow the speed of the car changes.
2
Simplify the function. Functions that are not simplified will still yield the same derivative, but it can be much more difficult to calculate.
- Example equation to simplify:
  - (6x + 8x)/2 +17x +4
  - (14x)/2 + 17x + 4
  - 7x + 17x + 4
  - 24x + 4
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3
Identify the form of the function. Learn the various forms.
- Just a number (e.g., 4)
- A number multiplied by a variable with no exponent (e.g., 4x)
- A number multiplied by a variable with an exponent (e.g., 4x^2)
- Addition (e.g., 4x + 4)
- Multiplication of variables (e.g., of the form x*x)
- Division of variables (e.g., of the form x/x)

Part 2

Part 2 of 2:

Finding the Derivates of Different Forms

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1
A number: The derivative of a function of this form is always zero. This is because there is no change in the function—the value of the function will always be the number that you're given. Here are some examples:
- (4)' = 0
- (-234059)' = 0
- (pi)' = 0
2
A number multiplied by a variable with no exponent: The derivative of a function of this form is always the number. If x does not have an exponent, the function is growing at a constant, steady, unchanging rate. You may recognize this trick from the linear equation y = mx + b. Check out these examples:
- (4x)' = 4
- (x)' = 1
- (-23x)' = -23
3
A number multiplied by a variable with an exponent: Subtract one from the exponent. Multiply the number by the value of the exponent. For instance:
- - (4x^3)' = (4*3)(x^(3-1)) = 12x^2
  - (2x^7)' = 14x^6
  - (3x^(-1))' = -3x^(-2)
4
Addition: Take the derivative of each part of the expression separately. For example:
- (4x + 4)' = 4 + 0 = 4
- ((x^2) + 7x)' = 2x + 7
5
Multiplication of variables: Multiply the first variable by the derivative of the second variable. Multiply the second variable by the derivative of the first variable. Add your two results together. Here's an example:
- ((x^2)*x)' = (x^2)*1 + x*2x = (x^2) + 2x*x = 3x^2
6
Division of variables: Multiply the bottom variable by the derivative of the top variable. Multiply the top variable by the derivative of the bottom variable. Subtract your result in Step 2 from your result in Step 1. Be careful, order matters! Divide your result in Step 3 with the square of the bottom variable. Check out this example:
- ((x^7)/x)' = (7x^6*x – 1*x^7)/(x^2) = (7x^7 - x^7)/(x^2) = 6x^7/x^2 = 6x^5
  - This is perhaps the hardest of the tricks to do, but it's well worth the effort. Make sure to do the steps in order and subtract in the correct order, and this will go smoothly.

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How do I know if a piecewise function is continuous?
Jake Adams
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Jake Adams is an academic tutor and the owner of Simplifi EDU, a Santa Monica, California based online tutoring business offering learning resources and online tutors for academic subjects K-College, SAT & ACT prep, and college admissions applications. With over 14 years of professional tutoring experience, Jake is dedicated to providing his clients the very best online tutoring experience and access to a network of excellent undergraduate and graduate-level tutors from top colleges all over the nation. Jake holds a BS in International Business and Marketing from Pepperdine University.
Jake Adams
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Expert Answer
In the context of a piecewise function, continuity is achieved when, from both the right and left approaches, the function values (f of X or Y) coincide at a specific X value. In simpler terms, the functions smoothly connect, and there is mutual agreement that a particular X value yields the same result for both functions. However, the differentiability of the piecewise function is contingent on whether the derivatives concur in terms of the values approached from both sides.
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What is the derivative of y=x^2 -2x+4?
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For a polynomial like this, the derivative of the function is equal to the derivative of each term individually, then added together. The derivative of x^2 is 2x. The derivative of -2x is -2. The derivative of any constant number, such as 4, is 0. Put these together, and the derivative of this function is 2x-2.
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How can I find the function if I know its derivative?
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To find the original function again, you would integrate the function, which is the opposite of differentiating it.
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This guide is meant to provide one with the tools one will need to calculate derivatives of basic functions. For an in-depth view of derivatives or for more advanced forms of differentiation such as the chain rule or partial differentiation, consulting the text Calculus: Early Transcendentals by James Stewart is recommended.
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