Matemáticas 5 · Tema 1
Reglas básicas de derivación
Videos
Video 1 — Reglas básicas
Video 2 — Con x en el denominador
Apuntes
Derivada de una constante
$f(x) = c \;\rightarrow\; f'(x) = 0$
Derivada de una función lineal
$f(x) = ax \;\rightarrow\; f'(x) = a$
Regla de la potencia
$f(x) = ax^n \;\rightarrow\; f'(x) = anx^{n-1}$
Regla de la suma y resta
$f = f_1 + f_2 + f_3 \;\rightarrow\; f' = f_1' + f_2' + f_3'$
Ejemplo
Deriva las siguientes funciones aplicando las reglas básicas:
$$\begin{aligned}
f(x) &= 5 \\
\mathbf{f'(x)} &= \mathbf{0}
\end{aligned}$$
$$\begin{aligned}
g(x) &= -3x \\
\mathbf{g'(x)} &= \mathbf{-3}
\end{aligned}$$
$$\begin{aligned}
h(x) &= 5x^4 - 7x^3 + 3x^2 - 2x + 7 \\
h'(x) &= 5(4)x^{4-1} - 7(3)x^{3-1} + 3(2)x^{2-1} - 2 \\
\mathbf{h'(x)} &= \mathbf{20x^3 - 21x^2 + 6x - 2}
\end{aligned}$$
$$\begin{aligned}
f(x) &= \frac{3}{x^4} \\
f(x) &= 3x^{-4} \\
f'(x) &= 3(-4)x^{-4-1} \\
f'(x) &= -12x^{-5} \\
\mathbf{f'(x)} &= \mathbf{-\frac{12}{x^5}}
\end{aligned}$$
Ejercicios
Bloque 1
$f(x) = 5x^6 - 3x^4 + 2x^3 - 5x - 4$
$f(x) = -4x^8 - 6x^5 + 3x^4 - 2x^3 + x + 2$
$f(x) = 3x^5 + 2x^4 - 7x^3 + 5x^2 - 4x$
$f(x) = 6x^4 - 5x^3 + 4x^2 - 6x - 1$
Bloque 2
$f(x) = \dfrac{5}{x^2} + \dfrac{3}{x^4} - \dfrac{2}{x^7}$
$f(x) = -\dfrac{6}{x^4} - \dfrac{3}{x^5} + \dfrac{7}{x^2}$
$f(x) = -\dfrac{1}{x^6} + \dfrac{4}{x^3} - \dfrac{2}{x^5}$
$f(x) = -\dfrac{2}{x^8} - \dfrac{3}{x^5} + \dfrac{8}{x^4}$
Bloque 3
$f(x) = 3x^4 - 2x + \dfrac{5}{x^2} - 7$
$f(x) = -x^5 + \dfrac{1}{x^4} + 6x - 12$
$f(x) = \dfrac{4}{x^3} - 2x^2 + 5x + 3$
$f(x) = 6x^6 - x + \dfrac{2}{x^5} + 9$
Bloque 4
$f(x) = -\dfrac{1}{x^8} + 5x^3 - 4x + 2$
$f(x) = 2x^7 - \dfrac{3}{x^5} + x^2 - 10$
$f(x) = 4x^4 - 6x^2 + \dfrac{8}{x} - 5$
$f(x) = -\dfrac{2}{x^6} + 3x^5 - 7x + 1$
Respuestas
▼Bloque 1
$$\begin{aligned}
f(x) &= 5x^6 - 3x^4 + 2x^3 - 5x - 4 \\
f'(x) &= 5(6)x^{5} - 3(4)x^{3} + 2(3)x^{2} - 5 \\
\mathbf{f'(x)} &= \mathbf{30x^5 - 12x^3 + 6x^2 - 5}
\end{aligned}$$
$$\begin{aligned}
f(x) &= -4x^8 - 6x^5 + 3x^4 - 2x^3 + x + 2 \\
f'(x) &= -4(8)x^7 - 6(5)x^4 + 3(4)x^3 - 2(3)x^2 + 1 \\
\mathbf{f'(x)} &= \mathbf{-32x^7 - 30x^4 + 12x^3 - 6x^2 + 1}
\end{aligned}$$
$$\begin{aligned}
f(x) &= 3x^5 + 2x^4 - 7x^3 + 5x^2 - 4x \\
f'(x) &= 3(5)x^4 + 2(4)x^3 - 7(3)x^2 + 5(2)x - 4 \\
\mathbf{f'(x)} &= \mathbf{15x^4 + 8x^3 - 21x^2 + 10x - 4}
\end{aligned}$$
$$\begin{aligned}
f(x) &= 6x^4 - 5x^3 + 4x^2 - 6x - 1 \\
f'(x) &= 6(4)x^3 - 5(3)x^2 + 4(2)x - 6 \\
\mathbf{f'(x)} &= \mathbf{24x^3 - 15x^2 + 8x - 6}
\end{aligned}$$
Bloque 2
$$\begin{aligned}
f(x) &= \frac{5}{x^2} + \frac{3}{x^4} - \frac{2}{x^7} \\
f(x) &= 5x^{-2} + 3x^{-4} - 2x^{-7} \\
f'(x) &= 5(-2)x^{-3} + 3(-4)x^{-5} - 2(-7)x^{-8} \\
f'(x) &= -10x^{-3} - 12x^{-5} + 14x^{-8} \\
\mathbf{f'(x)} &= \mathbf{-\frac{10}{x^3} - \frac{12}{x^5} + \frac{14}{x^8}}
\end{aligned}$$
$$\begin{aligned}
f(x) &= -\frac{6}{x^4} - \frac{3}{x^5} + \frac{7}{x^2} \\
f(x) &= -6x^{-4} - 3x^{-5} + 7x^{-2} \\
f'(x) &= -6(-4)x^{-5} - 3(-5)x^{-6} + 7(-2)x^{-3} \\
f'(x) &= 24x^{-5} + 15x^{-6} - 14x^{-3} \\
\mathbf{f'(x)} &= \mathbf{\frac{24}{x^5} + \frac{15}{x^6} - \frac{14}{x^3}}
\end{aligned}$$
$$\begin{aligned}
f(x) &= -\frac{1}{x^6} + \frac{4}{x^3} - \frac{2}{x^5} \\
f(x) &= -x^{-6} + 4x^{-3} - 2x^{-5} \\
f'(x) &= -(-6)x^{-7} + 4(-3)x^{-4} - 2(-5)x^{-6} \\
f'(x) &= 6x^{-7} - 12x^{-4} + 10x^{-6} \\
\mathbf{f'(x)} &= \mathbf{\frac{6}{x^7} - \frac{12}{x^4} + \frac{10}{x^6}}
\end{aligned}$$
$$\begin{aligned}
f(x) &= -\frac{2}{x^8} - \frac{3}{x^5} + \frac{8}{x^4} \\
f(x) &= -2x^{-8} - 3x^{-5} + 8x^{-4} \\
f'(x) &= -2(-8)x^{-9} - 3(-5)x^{-6} + 8(-4)x^{-5} \\
f'(x) &= 16x^{-9} + 15x^{-6} - 32x^{-5} \\
\mathbf{f'(x)} &= \mathbf{\frac{16}{x^9} + \frac{15}{x^6} - \frac{32}{x^5}}
\end{aligned}$$
Bloque 3
$$\begin{aligned}
f(x) &= 3x^4 - 2x + \frac{5}{x^2} - 7 \\
f(x) &= 3x^4 - 2x + 5x^{-2} - 7 \\
f'(x) &= 3(4)x^3 - 2 + 5(-2)x^{-3} \\
f'(x) &= 12x^3 - 2 - 10x^{-3} \\
\mathbf{f'(x)} &= \mathbf{12x^3 - 2 - \frac{10}{x^3}}
\end{aligned}$$
$$\begin{aligned}
f(x) &= -x^5 + \frac{1}{x^4} + 6x - 12 \\
f(x) &= -x^5 + x^{-4} + 6x - 12 \\
f'(x) &= -5x^4 - 4x^{-5} + 6 \\
\mathbf{f'(x)} &= \mathbf{-5x^4 - \frac{4}{x^5} + 6}
\end{aligned}$$
$$\begin{aligned}
f(x) &= \frac{4}{x^3} - 2x^2 + 5x + 3 \\
f(x) &= 4x^{-3} - 2x^2 + 5x + 3 \\
f'(x) &= 4(-3)x^{-4} - 2(2)x + 5 \\
f'(x) &= -12x^{-4} - 4x + 5 \\
\mathbf{f'(x)} &= \mathbf{-\frac{12}{x^4} - 4x + 5}
\end{aligned}$$
$$\begin{aligned}
f(x) &= 6x^6 - x + \frac{2}{x^5} + 9 \\
f(x) &= 6x^6 - x + 2x^{-5} + 9 \\
f'(x) &= 6(6)x^5 - 1 + 2(-5)x^{-6} \\
f'(x) &= 36x^5 - 1 - 10x^{-6} \\
\mathbf{f'(x)} &= \mathbf{36x^5 - 1 - \frac{10}{x^6}}
\end{aligned}$$
Bloque 4
$$\begin{aligned}
f(x) &= -\frac{1}{x^8} + 5x^3 - 4x + 2 \\
f(x) &= -x^{-8} + 5x^3 - 4x + 2 \\
f'(x) &= -(-8)x^{-9} + 5(3)x^2 - 4 \\
f'(x) &= 8x^{-9} + 15x^2 - 4 \\
\mathbf{f'(x)} &= \mathbf{\frac{8}{x^9} + 15x^2 - 4}
\end{aligned}$$
$$\begin{aligned}
f(x) &= 2x^7 - \frac{3}{x^5} + x^2 - 10 \\
f(x) &= 2x^7 - 3x^{-5} + x^2 - 10 \\
f'(x) &= 2(7)x^6 - 3(-5)x^{-6} + 2x \\
f'(x) &= 14x^6 + 15x^{-6} + 2x \\
\mathbf{f'(x)} &= \mathbf{14x^6 + \frac{15}{x^6} + 2x}
\end{aligned}$$
$$\begin{aligned}
f(x) &= 4x^4 - 6x^2 + \frac{8}{x} - 5 \\
f(x) &= 4x^4 - 6x^2 + 8x^{-1} - 5 \\
f'(x) &= 4(4)x^3 - 6(2)x + 8(-1)x^{-2} \\
f'(x) &= 16x^3 - 12x - 8x^{-2} \\
\mathbf{f'(x)} &= \mathbf{16x^3 - 12x - \frac{8}{x^2}}
\end{aligned}$$
$$\begin{aligned}
f(x) &= -\frac{2}{x^6} + 3x^5 - 7x + 1 \\
f(x) &= -2x^{-6} + 3x^5 - 7x + 1 \\
f'(x) &= -2(-6)x^{-7} + 3(5)x^4 - 7 \\
f'(x) &= 12x^{-7} + 15x^4 - 7 \\
\mathbf{f'(x)} &= \mathbf{\frac{12}{x^7} + 15x^4 - 7}
\end{aligned}$$
Ejercicios extra
▼Bloque 1
$f(x) = 8x^5 - 4x^3 + 2x^2 - 9$
$f(x) = -3x^4 + 5x^2 - 7x + 10$
$f(x) = 10x^3 - 7x^2 + 6x - 12$
Bloque 2
$f(x) = \dfrac{7}{x^3} - \dfrac{2}{x^6} + \dfrac{4}{x}$
$f(x) = -\dfrac{5}{x^7} + \dfrac{3}{x^2} - \dfrac{1}{x^5}$
$f(x) = \dfrac{4}{x^4} - \dfrac{3}{x^5} + \dfrac{9}{x^2}$
Bloque 3
$f(x) = -x^9 + 4x^5 - 2x^3 + x - 5$
$f(x) = \dfrac{1}{x^{10}} + \dfrac{1}{x^2} - 6x^4$
$f(x) = 2x^7 - 3x^4 + \dfrac{5}{x^3} + 12$
Bloque 4
$f(x) = -6x^2 + \dfrac{9}{x^2} - \dfrac{2}{x^8} + 3$
$f(x) = 5x^4 - 2x^3 + 7x^2 - 4x + 1$
$f(x) = -\dfrac{3}{x^6} - 4x^3 + \dfrac{2}{x^4} - 11$