AcademyDifferential Calculus
Academy
Differentiation Rules
Level 1 - Calculus topic page in Differential Calculus.
Principle
Differentiation rules let you find derivatives quickly once the basic derivative patterns are known. They preserve the meaning of derivative as rate of change while avoiding repeated limit calculations.
Notation
\(\frac{d}{dx}\)
differentiate with respect to x
\(n\)
constant exponent
\(c\)
constant scalar
The Core Method
Use the constant, sum, scalar multiple, and power rules.
Power rule
\[\frac{d}{dx}x^n=nx^{n-1}\]
Linearity
\[\frac{d}{dx}(af(x)+bg(x))=af'(x)+bg'(x)\]
Apply rules term-by-term, then simplify.
Worked Cases
Question
Find \(\frac{d}{dx}(4x^5-3x^2+7)\).
Answer
Differentiate each term. The derivative of \(4x^5\) is \(20x^4\). The derivative of \(-3x^2\) is \(-6x\). The derivative of the constant \(7\) is \(0\). Therefore the derivative is \(20x^4-6x\).
Examples
Question
Differentiate \(f(x)=x^{-2}+5x\).
Answer
Using the power rule, \(\frac{d}{dx}x^{-2}=-2x^{-3}\), and \(\frac{d}{dx}5x=5\). Hence \(f'(x)=-2x^{-3}+5\).