Derivatives of Other Elementary Functions

INTRODUCTION TO A WIDER RANGE OF FUNCTIONS

In analysis, we deal with several types of basic functions that appear when finding derivatives. Besides polynomials, important functions include exponential, logarithmic, trigonometric functions, and their inverse forms. These are called elementary functions because they represent the building blocks of most more complex expressions. Each of them has its own rule for differentiation, based on the mathematical properties of the function.

EXPONENTIAL FUNCTIONS

For a function of the form f(x) = a^x, where 'a' is a positive number different from 1, the derivative is given as:
f′(x) = a^x * ln(a).

A special case is the function f(x) = e^x, where e ≈ 2.718 (Euler's number), for which:
f′(x) = e^x.
This function remains unchanged upon differentiation.

LOGARITHMIC FUNCTIONS

The function f(x) = log_a(x), where a > 0 and a ≠ 1, has the derivative:
f′(x) = 1 / (x * ln(a)).

For the natural logarithm, i.e., f(x) = ln(x), a simplified rule applies:
f′(x) = 1 / x, valid for x > 0.
Logarithmic functions are not defined for negative values of x or for x = 0.

TRIGONOMETRIC FUNCTIONS

The derivatives of the basic trigonometric functions are:

• d/dx (sin(x)) = cos(x)
• d/dx (cos(x)) = -sin(x)
• d/dx (tan(x)) = 1 / cos²(x) = sec²(x), for x ≠ (2k + 1)π/2 (where k is an integer)
• d/dx (cot(x)) = -1 / sin²(x) = -csc²(x), for x ≠ kπ (where k is an integer)

The tangent and cotangent functions have points where they are not defined, so their derivatives do not exist there.

INVERSE TRIGONOMETRIC FUNCTIONS

The inverse trigonometric functions have the following derivatives:

• d/dx (arcsin(x)) = 1 / sqrt(1 − x²), for |x| < 1
• d/dx (arccos(x)) = −1 / sqrt(1 − x²), for |x| < 1
• d/dx (arctan(x)) = 1 / (1 + x²), for all real x

These functions are used when solving problems involving compositions of functions or when there is a need to invert trigonometric expressions.

CONCLUSION

Different types of elementary functions require knowledge of their specific differentiation rules. Each type of function has its prescribed rule, based on the structure of the function and its domain. Understanding these rules is essential for more complex treatment of expressions in analysis.