Calcular Derivadas derivada de una funciónEjercicios_Resueltos Calcula la derivada de las funciones: – – – – – – SOLUCIÓN Comentar el ejercicio
![\displaystyle \begin{array}{rcl@{\quad}l}f(x) & = & \mathrm{sen}\left(x\right) & \\[4pt] f^{\prime}(x) & = & \left(\mathrm{sen}\left(x\right)\right)^{\prime} & \text{Definición} \\[4pt] f^{\prime}(x) & = & \cos\left(x\right) \cdot \left(x\right)^{\prime} & \text{Derivada del seno} \\[4pt] f^{\prime}(x) & = & \cos\left(x\right) \cdot 1 & u^{\prime} = 1 \\[4pt] f^{\prime}(x) & = & \cos\left(x\right) & \text{Simplificamos}\end{array}](local/cache-TeX/bd7413f535722d7af52e345e0203f25e.png "\displaystyle \begin{array}{rcl@{\quad}l}f(x) & = & \mathrm{sen}\left(x\right) & \\[4pt] f^{\prime}(x) & = & \left(\mathrm{sen}\left(x\right)\right)^{\prime} & \text{Definición} \\[4pt] f^{\prime}(x) & = & \cos\left(x\right) \cdot \left(x\right)^{\prime} & \text{Derivada del seno} \\[4pt] f^{\prime}(x) & = & \cos\left(x\right) \cdot 1 & u^{\prime} = 1 \\[4pt] f^{\prime}(x) & = & \cos\left(x\right) & \text{Simplificamos}\end{array}")
![\displaystyle \begin{array}{rcl@{\quad}l}f(x) & = & \mathrm{sen}^{3}\left(x\right) & \\[4pt] f^{\prime}(x) & = & \left(\mathrm{sen}^{3}\left(x\right)\right)^{\prime} & \text{Definición} \\[4pt] f^{\prime}(x) & = & 3\mathrm{sen}^{3 - 1}\left(x\right) \cdot \left(\mathrm{sen}\left(x\right)\right)^{\prime} & \text{Derivada de una potencia (regla de la cadena)} \\[4pt] f^{\prime}(x) & = & 3\mathrm{sen}^{3 - 1}\left(x\right) \cdot \cos\left(x\right) & u^{\prime} = \cos\left(x\right) \\[4pt] f^{\prime}(x) & = & 3\mathrm{sen}^{2}\left(x\right) \cdot \cos\left(x\right) & \text{Simplificamos}\end{array}](local/cache-TeX/fd1530db1261753dc4628c119b3acfd8.png "\displaystyle \begin{array}{rcl@{\quad}l}f(x) & = & \mathrm{sen}^{3}\left(x\right) & \\[4pt] f^{\prime}(x) & = & \left(\mathrm{sen}^{3}\left(x\right)\right)^{\prime} & \text{Definición} \\[4pt] f^{\prime}(x) & = & 3\mathrm{sen}^{3 - 1}\left(x\right) \cdot \left(\mathrm{sen}\left(x\right)\right)^{\prime} & \text{Derivada de una potencia (regla de la cadena)} \\[4pt] f^{\prime}(x) & = & 3\mathrm{sen}^{3 - 1}\left(x\right) \cdot \cos\left(x\right) & u^{\prime} = \cos\left(x\right) \\[4pt] f^{\prime}(x) & = & 3\mathrm{sen}^{2}\left(x\right) \cdot \cos\left(x\right) & \text{Simplificamos}\end{array}")
![\displaystyle \begin{array}{l}\begin{array}{rcl@{\quad}l}f(x) & = & \left(\mathrm{sen}^{3}\left(x\right)\right)^{3} & \\[4pt] f^{\prime}(x) & = & \left(\left(\mathrm{sen}^{3}\left(x\right)\right)^{3}\right)^{\prime} & \text{Definición} \\[4pt] f^{\prime}(x) & = & 3\left(\mathrm{sen}^{3}\left(x\right)\right)^{3 - 1} \cdot 3\mathrm{sen}^{2}\left(x\right) \cdot \cos\left(x\right) & \text{Derivada de una potencia (regla de la cadena)} \\[4pt] f^{\prime}(x) & = & 3\mathrm{sen}^{6}\left(x\right) \cdot 3\mathrm{sen}^{2}\left(x\right) \cdot \cos\left(x\right) & \text{Simplificamos}\end{array} \\[20pt]\text{Cálculo auxiliar de }u^{\prime}: \\[2pt]\quad \begin{array}{rcl@{\quad}l}u & = & \mathrm{sen}^{3}\left(x\right) & \\[4pt] u^{\prime} & = & \left(\mathrm{sen}^{3}\left(x\right)\right)^{\prime} & \text{Definición} \\[4pt] u^{\prime} & = & 3\mathrm{sen}^{3 - 1}\left(x\right) \cdot \left(\mathrm{sen}\left(x\right)\right)^{\prime} & \text{Derivada de una potencia (regla de la cadena)} \\[4pt] u^{\prime} & = & 3\mathrm{sen}^{3 - 1}\left(x\right) \cdot \cos\left(x\right) & u^{\prime} = \cos\left(x\right) \\[4pt] u^{\prime} & = & 3\mathrm{sen}^{2}\left(x\right) \cdot \cos\left(x\right) & \text{Simplificamos}\end{array}\end{array}](local/cache-TeX/b03122d4dc1709fba53ae2ad7a35d623.png "\displaystyle \begin{array}{l}\begin{array}{rcl@{\quad}l}f(x) & = & \left(\mathrm{sen}^{3}\left(x\right)\right)^{3} & \\[4pt] f^{\prime}(x) & = & \left(\left(\mathrm{sen}^{3}\left(x\right)\right)^{3}\right)^{\prime} & \text{Definición} \\[4pt] f^{\prime}(x) & = & 3\left(\mathrm{sen}^{3}\left(x\right)\right)^{3 - 1} \cdot 3\mathrm{sen}^{2}\left(x\right) \cdot \cos\left(x\right) & \text{Derivada de una potencia (regla de la cadena)} \\[4pt] f^{\prime}(x) & = & 3\mathrm{sen}^{6}\left(x\right) \cdot 3\mathrm{sen}^{2}\left(x\right) \cdot \cos\left(x\right) & \text{Simplificamos}\end{array} \\[20pt]\text{Cálculo auxiliar de }u^{\prime}: \\[2pt]\quad \begin{array}{rcl@{\quad}l}u & = & \mathrm{sen}^{3}\left(x\right) & \\[4pt] u^{\prime} & = & \left(\mathrm{sen}^{3}\left(x\right)\right)^{\prime} & \text{Definición} \\[4pt] u^{\prime} & = & 3\mathrm{sen}^{3 - 1}\left(x\right) \cdot \left(\mathrm{sen}\left(x\right)\right)^{\prime} & \text{Derivada de una potencia (regla de la cadena)} \\[4pt] u^{\prime} & = & 3\mathrm{sen}^{3 - 1}\left(x\right) \cdot \cos\left(x\right) & u^{\prime} = \cos\left(x\right) \\[4pt] u^{\prime} & = & 3\mathrm{sen}^{2}\left(x\right) \cdot \cos\left(x\right) & \text{Simplificamos}\end{array}\end{array}")
![\displaystyle \begin{array}{rcl@{\quad}l}f(x) & = & \mathrm{sen}\left(3x^{2} - x\right) & \\[4pt] f^{\prime}(x) & = & \left(\mathrm{sen}\left(3x^{2} - x\right)\right)^{\prime} & \text{Definición} \\[4pt] f^{\prime}(x) & = & \cos\left(3x^{2} - x\right) \cdot \left(3x^{2} - x\right)^{\prime} & \text{Derivada del seno} \\[4pt] f^{\prime}(x) & = & \cos\left(3x^{2} - x\right) \cdot \left(6x - 1\right) & u^{\prime} = 6x - 1\end{array}](local/cache-TeX/9a7102cc5e4faefd5ef344561f1a9424.png "\displaystyle \begin{array}{rcl@{\quad}l}f(x) & = & \mathrm{sen}\left(3x^{2} - x\right) & \\[4pt] f^{\prime}(x) & = & \left(\mathrm{sen}\left(3x^{2} - x\right)\right)^{\prime} & \text{Definición} \\[4pt] f^{\prime}(x) & = & \cos\left(3x^{2} - x\right) \cdot \left(3x^{2} - x\right)^{\prime} & \text{Derivada del seno} \\[4pt] f^{\prime}(x) & = & \cos\left(3x^{2} - x\right) \cdot \left(6x - 1\right) & u^{\prime} = 6x - 1\end{array}")
![\displaystyle \begin{array}{l}\begin{array}{rcl@{\quad}l}f(x) & = & \left(\mathrm{sen}^{4}\left(3x^{4} - 5x\right)\right)^{3} & \\[4pt] f^{\prime}(x) & = & \left(\left(\mathrm{sen}^{4}\left(3x^{4} - 5x\right)\right)^{3}\right)^{\prime} & \text{Definición} \\[4pt] f^{\prime}(x) & = & 3\left(\mathrm{sen}^{4}\left(3x^{4} - 5x\right)\right)^{3 - 1} \cdot 4\mathrm{sen}^{3}\left(3x^{4} - 5x\right) \cdot \cos\left(3x^{4} - 5x\right) \cdot \left(12x^{3} - 5\right) & \text{Derivada de una potencia (regla de la cadena)} \\[4pt] f^{\prime}(x) & = & 3\mathrm{sen}^{8}\left(3x^{4} - 5x\right) \cdot 4\mathrm{sen}^{3}\left(3x^{4} - 5x\right) \cdot \cos\left(3x^{4} - 5x\right) \cdot \left(12x^{3} - 5\right) & \text{Simplificamos}\end{array} \\[20pt]\text{Cálculo auxiliar de }u^{\prime}: \\[2pt]\quad \begin{array}{l}\begin{array}{rcl@{\quad}l}u & = & \mathrm{sen}^{4}\left(3x^{4} - 5x\right) & \\[4pt] u^{\prime} & = & \left(\mathrm{sen}^{4}\left(3x^{4} - 5x\right)\right)^{\prime} & \text{Definición} \\[4pt] u^{\prime} & = & 4\mathrm{sen}^{4 - 1}\left(3x^{4} - 5x\right) \cdot \cos\left(3x^{4} - 5x\right) \cdot \left(12x^{3} - 5\right) & \text{Derivada de una potencia (regla de la cadena)} \\[4pt] u^{\prime} & = & 4\mathrm{sen}^{3}\left(3x^{4} - 5x\right) \cdot \cos\left(3x^{4} - 5x\right) \cdot \left(12x^{3} - 5\right) & \text{Simplificamos}\end{array} \\[20pt]\text{Cálculo auxiliar de }u^{\prime}: \\[2pt]\quad \begin{array}{rcl@{\quad}l}u & = & \mathrm{sen}\left(3x^{4} - 5x\right) & \\[4pt] u^{\prime} & = & \left(\mathrm{sen}\left(3x^{4} - 5x\right)\right)^{\prime} & \text{Definición} \\[4pt] u^{\prime} & = & \cos\left(3x^{4} - 5x\right) \cdot \left(3x^{4} - 5x\right)^{\prime} & \text{Derivada del seno} \\[4pt] u^{\prime} & = & \cos\left(3x^{4} - 5x\right) \cdot \left(12x^{3} - 5\right) & u^{\prime} = 12x^{3} - 5\end{array}\end{array}\end{array}](local/cache-TeX/a0a4de734457207ab34327c19872f379.png "\displaystyle \begin{array}{l}\begin{array}{rcl@{\quad}l}f(x) & = & \left(\mathrm{sen}^{4}\left(3x^{4} - 5x\right)\right)^{3} & \\[4pt] f^{\prime}(x) & = & \left(\left(\mathrm{sen}^{4}\left(3x^{4} - 5x\right)\right)^{3}\right)^{\prime} & \text{Definición} \\[4pt] f^{\prime}(x) & = & 3\left(\mathrm{sen}^{4}\left(3x^{4} - 5x\right)\right)^{3 - 1} \cdot 4\mathrm{sen}^{3}\left(3x^{4} - 5x\right) \cdot \cos\left(3x^{4} - 5x\right) \cdot \left(12x^{3} - 5\right) & \text{Derivada de una potencia (regla de la cadena)} \\[4pt] f^{\prime}(x) & = & 3\mathrm{sen}^{8}\left(3x^{4} - 5x\right) \cdot 4\mathrm{sen}^{3}\left(3x^{4} - 5x\right) \cdot \cos\left(3x^{4} - 5x\right) \cdot \left(12x^{3} - 5\right) & \text{Simplificamos}\end{array} \\[20pt]\text{Cálculo auxiliar de }u^{\prime}: \\[2pt]\quad \begin{array}{l}\begin{array}{rcl@{\quad}l}u & = & \mathrm{sen}^{4}\left(3x^{4} - 5x\right) & \\[4pt] u^{\prime} & = & \left(\mathrm{sen}^{4}\left(3x^{4} - 5x\right)\right)^{\prime} & \text{Definición} \\[4pt] u^{\prime} & = & 4\mathrm{sen}^{4 - 1}\left(3x^{4} - 5x\right) \cdot \cos\left(3x^{4} - 5x\right) \cdot \left(12x^{3} - 5\right) & \text{Derivada de una potencia (regla de la cadena)} \\[4pt] u^{\prime} & = & 4\mathrm{sen}^{3}\left(3x^{4} - 5x\right) \cdot \cos\left(3x^{4} - 5x\right) \cdot \left(12x^{3} - 5\right) & \text{Simplificamos}\end{array} \\[20pt]\text{Cálculo auxiliar de }u^{\prime}: \\[2pt]\quad \begin{array}{rcl@{\quad}l}u & = & \mathrm{sen}\left(3x^{4} - 5x\right) & \\[4pt] u^{\prime} & = & \left(\mathrm{sen}\left(3x^{4} - 5x\right)\right)^{\prime} & \text{Definición} \\[4pt] u^{\prime} & = & \cos\left(3x^{4} - 5x\right) \cdot \left(3x^{4} - 5x\right)^{\prime} & \text{Derivada del seno} \\[4pt] u^{\prime} & = & \cos\left(3x^{4} - 5x\right) \cdot \left(12x^{3} - 5\right) & u^{\prime} = 12x^{3} - 5\end{array}\end{array}\end{array}")