Propiedades de la derivada#

Ultima modificación: Feb 01, 2024 | YouTube

Derivada de la multiplicación de una función por un escalar#

\begin{split} \frac{d}{dx} f(x) & = \frac{d}{dx} C \cdot g(x) \\ \\ & = C \cdot \frac{d}{dx} g(x) \\ \\ \end{split}

Ejemplo:

\begin{split} \frac{d}{dx} 5x^2 & = 5 \cdot \frac{d}{dx} x^2 \\ \\ & = 5 \cdot (2 \cdot x) \\ \\ & = 10 x \end{split}

Derivada de una suma de funciones#

\frac{d}{dx} \left[ f(x) + g(x) \right] = \frac{d}{dx} f(x) + \frac{d}{dx} g(x)

Ejemplo:

\begin{split} \frac{d}{dx} \left(5 x^2 + 3x + 2 \right) & = \frac{d}{dx} 5 x^2 + \frac{d}{dx} 3 x + \frac{d}{dx} 2 \\ \\ & = 5 \cdot \frac{d}{dx} x^2 + 3 \cdot \frac{d}{dx} x + 0 \\ \\ & = 5 \cdot (2 \cdot x) + 3 \cdot (1) + 0 \\ \\ & = 10 x + 3 \end{split}

Derivada del producto de funciones#

\frac{d}{dx} \left[ f(x) \cdot g(x) \right] = \left( f(x) \frac{d}{dx} g(x) \right) + \left( g(x) \frac{d}{dx} f(x) \right)

Ejemplo:

\begin{split} \frac{d}{dx} \left[ (4 x^2) \cdot (3x) \right] & = 4 x^2 \cdot \frac{d}{dx} (3x) + 3x \cdot \frac{d}{dx} (4x^2) \\ \\ & = 4 x^2 \cdot (3) + 3x \cdot (8 x) \\ \\ & = 12 x^2 + 24 x^2 \\ \\ & = 36 x^2 \end{split}

Chequeo:

\begin{split} \frac{d}{dx} \left[ (4 x^2) \cdot (3x) \right] & = \frac{d}{dx} \left( 12 x^3 \right) & = 12 \frac{d}{dx} x^3 \\ \\ & = 12 \cdot 3 \cdot x^{3-1} \\ \\ & = 36 x^2 \end{split}

Derivación en cadena#

y=f(u)

u = g(x)

entonces:

y=f(g(x))

\begin{split} \frac{dy}{dx} & = \frac{dy}{du} \times \frac{du}{dx} \\ \\ & = \frac{d}{du} f(u) \times \frac{d}{dx} g(x)\\ \\ \end{split}

Ejemplo: Calcule la derivada respecto a x de:

\begin{split} \sigma(x) & = \frac{1}{1 + \exp(-x)} \\ \\ & = \left[1 + \exp(-x) \right]^{-1} \\ \end{split}

En este caso:

\begin{split} y & = f(u) \\ \\ & = u^{-1} \\ \\ u & = g(x) \\ \\ & = 1 + \exp(-x) \end{split}

Entonces:

\begin{split} \frac{d}{dx} \sigma (x) & = \frac{d}{du} \left( u^{-1} \right) \times \frac{d}{dx} \left[ 1 + \exp(-x) \right] \\ \\ & = \left( - u^{-2} \right) \times \left[ \frac{d}{dx} (1) + \frac{d}{dx} \exp(-x) \right] \\ \\ & = - \frac{1}{\left[ 1+\exp(-x) \right]^2} \times \left[ - \exp(-x) \right] \\ \\ & = \frac{ \exp(-x) }{\left[ 1+\exp(-x) \right]^2} \\ \\ & = \frac{1}{1+\exp(-x)} \times \frac{\exp(x)}{1+\exp(-x)} \\ \\ & = \frac{1}{1+\exp(-x)} \times \frac{\exp(x) + 1 - 1}{1+\exp(-x)} \\ \\ & = \frac{1}{1+\exp(-x)} \times \left[\frac{1+\exp(-x)}{1+\exp(-x)} - \frac{1}{1+\exp(-x)} \right] \\ \\ & = \sigma(x) \left[ 1 - \sigma(x) \right] \\ \end{split}