Calculus
Essential formulas and concepts in calculus
Definition of Derivative
\(f'(x) = \displaystyle \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\)
Variables:
- \(f(x) is a function, f'(x) is its derivative. h is an increment in x (approaching 0 in the limit).\)
Description/Usage:
Defines the derivative as the instantaneous rate of change or slope of the function \(f(x)\) at point \(x\). Graphically, it’s the slope of the tangent line to \(y=f(x)\) at \(x\).
Mathematical concept visualization
Power Rule (for derivatives)
\(\displaystyle \frac{d}{dx}\big(x^n\big) = n x^{n-1}\)
Variables:
- \(n is a constant exponent, x is the variable.\)
Description/Usage:
Provides a quick way to differentiate monomials. For example, if \(f(x)=x^5\), then \(f'(x)=5x^4\). This rule is fundamental for polynomial differentiation.
Mathematical concept visualization
Indefinite Integral (Power Rule)
\(\displaystyle \int x^n\,dx = \frac{x^{n+1}}{n+1} + C \qquad(n \neq -1)\)
Variables:
- \(n is a constant exponent (not -1); C is the constant of integration (since antiderivatives are determined up to an additive constant).\)
Description/Usage:
Mathematical concept visualization
Integration by Parts
\(\displaystyle \int u\,dv = uv - \int v\,du\)
Variables:
- \(u and v are functions of x such that du and dv are their differentials (derivatives times dx). Often one sets u=f(x) and dv = g(x)dx.\)
Description/Usage:
Mathematical concept visualization