Differentiation Formulas for K-12 Students
Differentiation is a fundamental concept in calculus that helps us find rates of change and slopes of curves. Below are the essential differentiation formulas that K-12 students should know, organized by category.
Each formula is accompanied by a simple example to illustrate its application.
Basic Differentiation Rules
Constant Rule
\[ \frac{d}{dx}(c) = 0 \]
Example:
\[ \frac{d}{dx}(7) = 0 \]Power Rule
\[ \frac{d}{dx}(x^n) = n \cdot x^{n-1} \]
Example:
\[ \frac{d}{dx}(x^3) = 3x^2 \]Constant Multiple Rule
\[ \frac{d}{dx}(c \cdot f(x)) = c \cdot \frac{d}{dx}(f(x)) \]
Example:
\[ \frac{d}{dx}(5x^2) = 5 \cdot \frac{d}{dx}(x^2) = 5 \cdot 2x = 10x \]Sum Rule
\[ \frac{d}{dx}(f(x) + g(x)) = \frac{d}{dx}(f(x)) + \frac{d}{dx}(g(x)) \]
Example:
\[ \frac{d}{dx}(x^2 + x) = \frac{d}{dx}(x^2) + \frac{d}{dx}(x) = 2x + 1 \]Difference Rule
\[ \frac{d}{dx}(f(x) - g(x)) = \frac{d}{dx}(f(x)) - \frac{d}{dx}(g(x)) \]
Example:
\[ \frac{d}{dx}(x^3 - x) = \frac{d}{dx}(x^3) - \frac{d}{dx}(x) = 3x^2 - 1 \]Product and Quotient Rules
Product Rule
\[ \frac{d}{dx}(f(x) \cdot g(x)) = f(x) \cdot \frac{d}{dx}(g(x)) + g(x) \cdot \frac{d}{dx}(f(x)) \]
Example:
\[ \frac{d}{dx}(x^2 \cdot x^3) = x^2 \cdot \frac{d}{dx}(x^3) + x^3 \cdot \frac{d}{dx}(x^2) = x^2 \cdot 3x^2 + x^3 \cdot 2x = 3x^4 + 2x^4 = 5x^4 \]Quotient Rule
\[ \frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{g(x) \cdot \frac{d}{dx}(f(x)) - f(x) \cdot \frac{d}{dx}(g(x))}{[g(x)]^2} \]
Example:
\[ \frac{d}{dx}\left(\frac{x^2}{x+1}\right) = \frac{(x+1) \cdot \frac{d}{dx}(x^2) - x^2 \cdot \frac{d}{dx}(x+1)}{(x+1)^2} = \frac{(x+1) \cdot 2x - x^2 \cdot 1}{(x+1)^2} = \frac{2x^2 + 2x - x^2}{(x+1)^2} = \frac{x^2 + 2x}{(x+1)^2} \]Chain Rule
Chain Rule
\[ \frac{d}{dx}(f(g(x))) = f'(g(x)) \cdot g'(x) \]
Example:
\[ \frac{d}{dx}((x^2+1)^3) = 3(x^2+1)^2 \cdot \frac{d}{dx}(x^2+1) = 3(x^2+1)^2 \cdot 2x = 6x(x^2+1)^2 \]Trigonometric Functions
Sine
\[ \frac{d}{dx}(\sin x) = \cos x \]
Example:
\[ \frac{d}{dx}(\sin(2x)) = \cos(2x) \cdot \frac{d}{dx}(2x) = 2\cos(2x) \]Cosine
\[ \frac{d}{dx}(\cos x) = -\sin x \]
Example:
\[ \frac{d}{dx}(\cos(x^2)) = -\sin(x^2) \cdot \frac{d}{dx}(x^2) = -\sin(x^2) \cdot 2x = -2x\sin(x^2) \]Tangent
\[ \frac{d}{dx}(\tan x) = \sec^2 x \]
Example:
\[ \frac{d}{dx}(\tan(3x)) = \sec^2(3x) \cdot \frac{d}{dx}(3x) = 3\sec^2(3x) \]Exponential and Logarithmic Functions
Natural Exponential
\[ \frac{d}{dx}(e^x) = e^x \]
Example:
\[ \frac{d}{dx}(e^{2x}) = e^{2x} \cdot \frac{d}{dx}(2x) = e^{2x} \cdot 2 = 2e^{2x} \]General Exponential
\[ \frac{d}{dx}(a^x) = a^x \ln(a) \]
Example:
\[ \frac{d}{dx}(2^x) = 2^x \ln(2) \]Natural Logarithm
\[ \frac{d}{dx}(\ln x) = \frac{1}{x} \]
Example:
\[ \frac{d}{dx}(\ln(x^2)) = \frac{1}{x^2} \cdot \frac{d}{dx}(x^2) = \frac{1}{x^2} \cdot 2x = \frac{2x}{x^2} = \frac{2}{x} \]Logarithm with Base a
\[ \frac{d}{dx}(\log_a x) = \frac{1}{x \ln a} \]
Example:
\[ \frac{d}{dx}(\log_{10} x) = \frac{1}{x \ln(10)} \]
Note: Remember that differentiation helps us find the rate of change of a function. These formulas are essential tools for solving problems involving slopes, velocities, rates, and optimization in calculus.