Aller au contenu principal

Basic Properties and Formulas

Basic Derivative Rules​

  • Constant Rule: The derivative of a constant is zero. ddxc=0\frac{d}{dx} c = 0
  • Constant Multiple Rule: The derivative of a constant multiplied by a function is the constant times the derivative of the function. ddx[cf(x)]=cf′(x)\frac{d}{dx} [c f(x)] = c f'(x)
  • Power Rule: The derivative of a variable raised to a power is the power multiplied by the variable raised to one less than the power. ddxxn=nxn−1\frac{d}{dx} x^n = nx^{n-1}

Derivative of Composite Functions​

Sum Rule​

The derivative of a sum of two functions is the sum of their derivatives.

ddx[f(x)+g(x)]=f′(x)+g′(x)\frac{d}{dx} [f(x) + g(x)] = f'(x) + g'(x)

Difference Rule​

The derivative of a difference between two functions is the difference of their derivatives.

ddx[f(x)−g(x)]=f′(x)−g′(x)\frac{d}{dx} [f(x) - g(x)] = f'(x) - g'(x)

Product Rule​

The derivative of the product of two functions is given by:

ddx[f(x)g(x)]=f′(x)g(x)+f(x)g′(x)\frac{d}{dx} [f(x) g(x)] = f'(x) g(x) + f(x) g'(x)

Quotient Rule​

The derivative of the quotient of two functions is:

ddx(f(x)g(x))=f′(x)g(x)−f(x)g′(x)[g(x)]2\frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}

Chain Rule​

Formal Definition​

For two functions gg and hh, with y=g(u)y = g(u) and u=h(x)u = h(x), the derivative of yy with respect to xx is:

dydx=dydu⋅dudx\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}

Lagrange Notation​

Using Lagrange's notation, where ′' denotes the derivative, for y=g(h(x))y = g(h(x)), the chain rule is expressed as:

y′(x)=g′(h(x))⋅h′(x)y'(x) = g'(h(x)) \cdot h'(x)

Generalization for Multiple Compositions​

For compositions involving multiple functions, such as f=f∘g∘hf = f \circ g \circ h, the chain rule extends as follows:

dfdt=dfdg⋅dgdh⋅dhdt\frac{df}{dt} = \frac{df}{dg} \cdot \frac{dg}{dh} \cdot \frac{dh}{dt}

Or, using Lagrange notation for a function f(t)f(t) that is a composition of ff, gg, and hh, we get:

(f∘g∘h)′(t)=f′(g(h(t)))⋅g′(h(t))⋅h′(t)(f \circ g \circ h)'(t) = f'(g(h(t))) \cdot g'(h(t)) \cdot h'(t)

Special Derivatives​

  • Derivative of a Square: The derivative of the square of a function is twice the function times the derivative of the function. ddx[f(x)]2=2f(x)f′(x)\frac{d}{dx} [f(x)]^2 = 2 f(x) f'(x)
  • Derivative of a Reciprocal: The derivative of the reciprocal of a function. ddx(1g(x))=−g′(x)[g(x)]2\frac{d}{dx} \left( \frac{1}{g(x)} \right) = -\frac{g'(x)}{[g(x)]^2}

References​