Differentiation Rules

Differentiation Rules

Constant Function Rule

  • The derivative of any constant is zero. f′(x)=dydx=0f'(x) = \frac{dy}{dx} = 0

Identity Function Rule

  • The derivative of f(x)=xf(x) = x is always 1. f′(x)=dydx=1f'(x) = \frac{dy}{dx} = 1

Power Rule

  • If f(x)=xnf(x) = x^n, then f′(x)=nxn−1f'(x) = nx^{n-1}

Constant Multiple Rule

  • The derivative of a constant C times a function is the constant times the derivative of the function: f′(x)=Cnxn−1f'(x) = Cnx^{n-1}

Linearity Rule

  • Applies when adding or subtracting functions multiplied by constants:
    ddx(u±v)=dudx±dvdx=u′(x)±v′(x)\frac{d}{dx}(u \pm v) = \frac{du}{dx} \pm \frac{dv}{dx} = u'(x) \pm v'(x)

Product Rule

  • The derivative is found by: f′(x)=u(x)∗v′(x)+v(x)∗u′(x)f'(x) = u(x) * v'(x) + v(x) * u'(x)

Quotient Rule

  • Applicable for the ratio of two differentiable functions u(x)u(x) and v(x)v(x): f′(x)=v(x)∗u′(x)−u(x)∗v′(x)[v(x)]2f'(x) = \frac{v(x) * u'(x) - u(x) * v'(x)}{[v(x)]^2}

Chain Rule

  • For functions raised to a certain power: Dx[f(x)]n=n[f(x)]n−1∗f′(x)D_x [f(x)]^n = n [f(x)]^{n-1} * f'(x)