Table of Derivatives of Simple Functions

Calculating derivatives is a fundamental operation in differential calculus. Below is a table for finding derivatives of simple functions.
For more complex differentiation rules, refer to other lessons:

• Table of derivatives of exponential and logarithmic functions
• Table of derivatives of trigonometric functions

These formulas will assist in solving differential equations and related problems.
The table of derivatives of simple functions includes a “cheat sheet” with the main cases for finding derivatives, along with explanations
for each case.

Derivatives of Simple Functions

The derivative of a constant is zero

1. c' = 0
   

   Example:

   5' = 0
   

   Explanation: The derivative represents the rate of change of a function’s value with respect to its argument.
   Since a constant does not change, its rate of change is always zero.

2. The derivative of a variable is one

   x' = 1
   

   Explanation: For each unit increase in the argument (x), the function (y = x) increases by the same amount.
   Thus, the rate of change of (y) with respect to (x) is one.

3. The derivative of a variable multiplied by a constant is the constant

   (cx)' = c
   

   Examples:

   (3x)' = 3
   

   (2x)' = 2
   

   Explanation: When the argument (x) changes, the function (y = cx) changes by (c) times the change in (x).
    Therefore, the rate of change of (y) with respect to (x) is (c). It follows that:

   (cx + b)' = c
   

   This means the derivative of the linear function (y = kx + b) is equal to the slope (k).

4. The derivative of the absolute value of a variable

   |x|' = x / |x|  for  х ≠ 0
   

   Explanation: The derivative of (x) is one. The derivative of (|x|) changes sign at the origin.
   For (x < 0), (|x| = -x), and for (x > 0), (|x| = x). Thus, x / |x| returns -1 for negative (x) and 1 for positive (x).

5. The derivative of a variable raised to a power

   ( x^c )'= cx^c-1  for  x^c  and  сx^c-1  defined and с ≠ 0
   

   Examples:

   (x^2)' = 2x
   

   (x^3)' = 3x^2
   

   Memory aid: Move the exponent down as a multiplier and decrease the exponent by one.
   For example, for (x^2), the 2 moves down, and the exponent decreases by one, giving (2x).
   Similarly, for (x^3), the 3 moves down, and the exponent decreases by one, giving (3x^2).

6. The derivative of a reciprocal function

   (1/х)' = - 1 / x²
   

   Explanation: Since (1/x)'  can be written as (x^-1 )', we apply the power rule:

   (x^-1 )' = -1x^-2 = - 1 / х²
   

7. The derivative of a reciprocal function with an arbitrary power

   ( 1 / x^c )' = - c / x^c+1
   

   Example:

   ( 1 / x² )' = - 2 / x³
   

8. The derivative of a square root function

   ( √x )' = 1 / ( 2√x )  or 1/2 х^-1/2
   

   Example:

   ( √x )' = ( х^1/2 )'
   

   Applying the power rule:

   ( х^1/2 )' = 1/2 х^-1/2 = 1 / (2√х)
   

9. The derivative of a root of an arbitrary degree

   ( ⁿ√x )' = 1 / ( n ⁿ√x^n-1 )
   

This table of derivatives covers the basic transformations that should be memorized.
For more complex derivatives, refer to the corresponding tables in other lessons.