1. Why Standard Derivatives Are Useful
Many functions appear again and again in calculus. Knowing their derivatives instantly makes differentiation faster and cleaner. These basic formulas act as building blocks for almost every derivative you compute later.
2. Polynomials
Polynomials are among the simplest functions to differentiate. The power rule handles every term easily.
2.1. Formula
\frac{d}{dx}(x^n) = n x^{n-1}
2.2. Example
\frac{d}{dx}(4x^3 - x^2 + 7) = 12x^2 - 2x
3. Rational Functions (Within Domain)
Rational functions use power rule and quotient rule wherever the function is defined.
3.1. Example
f(x) = \frac{1}{x} = x^{-1}
f'(x) = -x^{-2} = -\frac{1}{x^2}
4. Exponential Functions
Exponential functions are very smooth, and their derivatives are straightforward.
4.1. Formulas
\frac{d}{dx}(e^x) = e^x
\frac{d}{dx}(a^x) = a^x \ln(a) \quad (a > 0)
4.2. Example
\frac{d}{dx}(5^x) = 5^x \ln 5
5. Logarithmic Functions
Logarithmic functions grow slowly and have simple derivative rules.
5.1. Formulas
\frac{d}{dx}(\ln x) = \frac{1}{x}
\frac{d}{dx}(\log_a x) = \frac{1}{x \ln a}
5.2. Example
\frac{d}{dx}(\ln(x^2)) = \frac{2}{x}
6. Trigonometric Functions
Trigonometric derivatives are some of the most commonly used results in calculus.
6.1. Basic Formulas
\frac{d}{dx}(\sin x) = \cos x
\frac{d}{dx}(\cos x) = -\sin x
\frac{d}{dx}(\tan x) = \sec^2 x
\frac{d}{dx}(\cot x) = -\csc^2 x
\frac{d}{dx}(\sec x) = \sec x \tan x
\frac{d}{dx}(\csc x) = -\csc x \cot x
7. Inverse Trigonometric Functions
These functions appear often in integration and advanced calculus, so knowing their derivatives is very helpful.
7.1. Formulas
\frac{d}{dx}(\sin^{-1} x) = \frac{1}{\sqrt{1 - x^2}}
\frac{d}{dx}(\cos^{-1} x) = -\frac{1}{\sqrt{1 - x^2}}
\frac{d}{dx}(\tan^{-1} x) = \frac{1}{1 + x^2}
\frac{d}{dx}(\cot^{-1} x) = -\frac{1}{1 + x^2}
8. Root Functions
Root functions can be written using fractional powers and then differentiated with the power rule.
8.1. Example
f(x) = \sqrt{x} = x^{1/2}
f'(x) = \frac{1}{2} x^{-1/2} = \frac{1}{2\sqrt{x}}
9. Absolute Value Function
The absolute value function behaves differently on the left and right sides of 0, so it has no derivative at x = 0.
9.1. Derivative (Piecewise)
\frac{d}{dx}|x| = \begin{cases} 1, & x > 0 \\ -1, & x < 0 \\ \text{undefined}, & x = 0 \end{cases}
10. Summary
These basic derivatives form a toolkit for handling almost any function. More complicated derivatives are usually combinations of these standard forms along with product, quotient, and chain rules.