Derivatives of common functions

Most functions encountered in practice have known derivatives. Knowing these standard derivatives allows one to quickly find the derivative of more complex functions using the basic differentiation rules. This section covers the most important function types: polynomials, power functions, exponential functions, logarithms, and trigonometric functions.

 

 

Polynomials and power functions

The power rule is the foundation for almost all polynomials. If

 

$$ \large f(x) = x^n $$

 

then:

 

$$ \large f'(x) = n \cdot x^{n-1} $$

 

Examples:

 

$$ \large (x^3)' = 3x^2 \qquad (x^{-2})' = -2x^{-3} \qquad (\sqrt{x})' = \frac{1}{2\sqrt{x}} $$

 

If the polynomial consists of several terms, each term is differentiated separately. For example:

 

$$ \large f(x) = 2x^3 - 5x^2 + 3x - 7 \quad \Rightarrow \quad f'(x) = 6x^2 - 10x + 3 $$

 

 

Exponential functions

Exponential functions describe many natural processes such as growth, decay, and interest. For the natural exponential function:

 

$$ \large (e^x)' = e^x $$

 

This means that \( \large e^x \) is its own derivative — it changes at the same rate as its current value. For a general exponential function:

 

$$ \large (a^x)' = a^x \cdot \ln(a) $$

 

Example:

 

$$ \large (2^x)' = 2^x \cdot \ln(2) $$

 

If the exponent itself is a function, the chain rule is used. For example:

 

$$ \large (e^{3x})' = 3e^{3x} \qquad (2^{x^2})' = 2^{x^2} \cdot \ln(2) \cdot 2x $$

 

 

Logarithmic functions

The natural logarithmic function \( \large \ln(x) \) is the inverse of \( \large e^x \). Its derivative is:

 

$$ \large (\ln x)' = \frac{1}{x} $$

 

For a logarithm with another base \( \large a \):

 

$$ \large (\log_a x)' = \frac{1}{x \cdot \ln(a)} $$

 

Example:

 

$$ \large (\log_{10} x)' = \frac{1}{x \cdot \ln(10)} $$

 

 

Trigonometric functions

Trigonometric functions describe angles and circular motion. Their derivatives follow a fixed pattern:

 

$$ \large (\sin x)' = \cos x \qquad (\cos x)' = -\sin x \qquad (\tan x)' = \frac{1}{\cos^2 x} $$

 

Examples of composite functions:

 

$$ \large (\sin(2x))' = 2\cos(2x) \qquad (\cos(3x^2))' = -6x \sin(3x^2) $$

 

 

Combined functions

By combining these standard derivatives with the differentiation rules, even very complex functions can be handled. For example:

 

$$ \large f(x) = e^x \cdot \sin x \quad \Rightarrow \quad f'(x) = e^x \cdot \sin x + e^x \cdot \cos x = e^x(\sin x + \cos x) $$

 

Or with a logarithm and a power function:

 

$$ \large f(x) = x^2 \ln x \quad \Rightarrow \quad f'(x) = 2x \ln x + x $$

 

 

Summary

Table of the most common derivatives:

 

Function	Derivative
\( \large x^n \)	\( \large n \cdot x^{n-1} \)
\( \large e^x \)	\( \large e^x \)
\( \large a^x \)	\( \large a^x \cdot \ln(a) \)
\( \large \ln(x) \)	\( \large \frac{1}{x} \)
\( \large \sin(x) \)	\( \large \cos(x) \)
\( \large \cos(x) \)	\( \large -\sin(x) \)
\( \large \tan(x) \)	\( \large \frac{1}{\cos^2(x)} \)

 

 

These standard derivatives form the foundation of all further work in differential calculus. They are used to determine slopes, find extrema, solve optimization problems, and analyze curves. Once these are mastered, differentiating even complex functions becomes routine.