Inicio › Matemáticas 5 › Reglas básicas de derivación

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  →  f′(x)=0f(x) = c \;\rightarrow\; f'(x) = 0

Ejemplo

f(x)=5f′(x)=0\begin{aligned} f(x) &= 5 \\ \mathbf{f'(x)} &= \mathbf{0} \end{aligned}
Derivada de una función lineal
f(x)=ax  →  f′(x)=af(x) = ax \;\rightarrow\; f'(x) = a

Ejemplo

g(x)=−3xg′(x)=−3\begin{aligned} g(x) &= -3x \\ \mathbf{g'(x)} &= \mathbf{-3} \end{aligned}
Regla de la potencia
f(x)=axn  →  f′(x)=anxn−1f(x) = ax^n \;\rightarrow\; f'(x) = anx^{n-1}

Ejemplo

f(x)=4x5f′(x)=4(5)x5−1f′(x)=20x4\begin{aligned} f(x) &= 4x^5 \\ f'(x) &= 4(5)x^{5-1} \\ \mathbf{f'(x)} &= \mathbf{20x^4} \end{aligned}
f(x)=3x4f(x)=3x−4f′(x)=3(−4)x−4−1f′(x)=−12x−5f′(x)=−12x5\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}

Ejemplo

Deriva combinando varios términos a la vez — recuerda que la derivada de una suma es la suma de las derivadas: f=f1+f2+f3  →  f′=f1′+f2′+f3′f = f_1 + f_2 + f_3 \;\rightarrow\; f' = f_1' + f_2' + f_3'.

Sin x en el denominador

h(x)=5x4−7x3+3x2−2x+7h′(x)=5(4)x4−1−7(3)x3−1+3(2)x2−1−2h′(x)=20x3−21x2+6x−2\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}

Con x en el denominador

f(x)=2x3+6x−9+3x2−7x4f(x)=2x3+6x−9+3x−2−7x−4f′(x)=2(3)x2+6+3(−2)x−3−7(−4)x−5f′(x)=6x2+6−6x−3+28x−5f′(x)=6x2+6−6x3+28x5\begin{aligned} f(x) &= 2x^3 + 6x - 9 + \frac{3}{x^2} - \frac{7}{x^4} \\ f(x) &= 2x^3 + 6x - 9 + 3x^{-2} - 7x^{-4} \\ f'(x) &= 2(3)x^2 + 6 + 3(-2)x^{-3} - 7(-4)x^{-5} \\ f'(x) &= 6x^2 + 6 - 6x^{-3} + 28x^{-5} \\ \mathbf{f'(x)} &= \mathbf{6x^2 + 6 - \frac{6}{x^3} + \frac{28}{x^5}} \end{aligned}

Ejercicios

Bloque 1
f(x)=5x6−3x4+2x3−5x−4f(x) = 5x^6 - 3x^4 + 2x^3 - 5x - 4
f(x)=−4x8−6x5+3x4−2x3+x+2f(x) = -4x^8 - 6x^5 + 3x^4 - 2x^3 + x + 2
f(x)=3x5+2x4−7x3+5x2−4xf(x) = 3x^5 + 2x^4 - 7x^3 + 5x^2 - 4x
f(x)=6x4−5x3+4x2−6x−1f(x) = 6x^4 - 5x^3 + 4x^2 - 6x - 1
f(x)=−2x7+4x5−3x2+8x−6f(x) = -2x^7 + 4x^5 - 3x^2 + 8x - 6
f(x)=7x4−x3+5x2−9x+3f(x) = 7x^4 - x^3 + 5x^2 - 9x + 3
Bloque 2
f(x)=3x4−7−2x+5x2−4x3f(x) = 3x^4 - 7 - 2x + \dfrac{5}{x^2} - \dfrac{4}{x^3}
f(x)=1x4−x5+6x−12−3xf(x) = \dfrac{1}{x^4} - x^5 + 6x - 12 - \dfrac{3}{x}
f(x)=5x+4x3−2x2+3+6x5f(x) = 5x + \dfrac{4}{x^3} - 2x^2 + 3 + \dfrac{6}{x^5}
f(x)=9−x+6x6−7x2+2x5f(x) = 9 - x + 6x^6 - \dfrac{7}{x^2} + \dfrac{2}{x^5}
f(x)=−1x8+5x3+3x+2−4xf(x) = -\dfrac{1}{x^8} + 5x^3 + \dfrac{3}{x} + 2 - 4x
f(x)=2x7−10+8x3+4x−3x5f(x) = 2x^7 - 10 + \dfrac{8}{x^3} + 4x - \dfrac{3}{x^5}

Respuestas

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Bloque 1
f(x)=5x6−3x4+2x3−5x−4f′(x)=5(6)x5−3(4)x3+2(3)x2−5f′(x)=30x5−12x3+6x2−5\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}
f(x)=−4x8−6x5+3x4−2x3+x+2f′(x)=−4(8)x7−6(5)x4+3(4)x3−2(3)x2+1f′(x)=−32x7−30x4+12x3−6x2+1\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}
f(x)=3x5+2x4−7x3+5x2−4xf′(x)=3(5)x4+2(4)x3−7(3)x2+5(2)x−4f′(x)=15x4+8x3−21x2+10x−4\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}
f(x)=6x4−5x3+4x2−6x−1f′(x)=6(4)x3−5(3)x2+4(2)x−6f′(x)=24x3−15x2+8x−6\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}
f(x)=−2x7+4x5−3x2+8x−6f′(x)=−2(7)x6+4(5)x4−3(2)x+8f′(x)=−14x6+20x4−6x+8\begin{aligned} f(x) &= -2x^7 + 4x^5 - 3x^2 + 8x - 6 \\ f'(x) &= -2(7)x^6 + 4(5)x^4 - 3(2)x + 8 \\ \mathbf{f'(x)} &= \mathbf{-14x^6 + 20x^4 - 6x + 8} \end{aligned}
f(x)=7x4−x3+5x2−9x+3f′(x)=7(4)x3−3x2+5(2)x−9f′(x)=28x3−3x2+10x−9\begin{aligned} f(x) &= 7x^4 - x^3 + 5x^2 - 9x + 3 \\ f'(x) &= 7(4)x^3 - 3x^2 + 5(2)x - 9 \\ \mathbf{f'(x)} &= \mathbf{28x^3 - 3x^2 + 10x - 9} \end{aligned}
Bloque 2
f(x)=3x4−7−2x+5x2−4x3f(x)=3x4−7−2x+5x−2−4x−3f′(x)=3(4)x3−2+5(−2)x−3−4(−3)x−4f′(x)=12x3−2−10x−3+12x−4f′(x)=12x3−2−10x3+12x4\begin{aligned} f(x) &= 3x^4 - 7 - 2x + \frac{5}{x^2} - \frac{4}{x^3} \\ f(x) &= 3x^4 - 7 - 2x + 5x^{-2} - 4x^{-3} \\ f'(x) &= 3(4)x^3 - 2 + 5(-2)x^{-3} - 4(-3)x^{-4} \\ f'(x) &= 12x^3 - 2 - 10x^{-3} + 12x^{-4} \\ \mathbf{f'(x)} &= \mathbf{12x^3 - 2 - \frac{10}{x^3} + \frac{12}{x^4}} \end{aligned}
f(x)=1x4−x5+6x−12−3xf(x)=x−4−x5+6x−12−3x−1f′(x)=(−4)x−5−(5)x4+6−3(−1)x−2f′(x)=−4x−5−5x4+6+3x−2f′(x)=−4x5−5x4+6+3x2\begin{aligned} f(x) &= \frac{1}{x^4} - x^5 + 6x - 12 - \frac{3}{x} \\ f(x) &= x^{-4} - x^5 + 6x - 12 - 3x^{-1} \\ f'(x) &= (-4)x^{-5} - (5)x^4 + 6 - 3(-1)x^{-2} \\ f'(x) &= -4x^{-5} - 5x^4 + 6 + 3x^{-2} \\ \mathbf{f'(x)} &= \mathbf{-\frac{4}{x^5} - 5x^4 + 6 + \frac{3}{x^2}} \end{aligned}
f(x)=5x+4x3−2x2+3+6x5f(x)=5x+4x−3−2x2+3+6x−5f′(x)=5+4(−3)x−4−2(2)x+6(−5)x−6f′(x)=5−12x−4−4x−30x−6f′(x)=5−12x4−4x−30x6\begin{aligned} f(x) &= 5x + \frac{4}{x^3} - 2x^2 + 3 + \frac{6}{x^5} \\ f(x) &= 5x + 4x^{-3} - 2x^2 + 3 + 6x^{-5} \\ f'(x) &= 5 + 4(-3)x^{-4} - 2(2)x + 6(-5)x^{-6} \\ f'(x) &= 5 - 12x^{-4} - 4x - 30x^{-6} \\ \mathbf{f'(x)} &= \mathbf{5 - \frac{12}{x^4} - 4x - \frac{30}{x^6}} \end{aligned}
f(x)=9−x+6x6−7x2+2x5f(x)=9−x+6x6−7x−2+2x−5f′(x)=−1+6(6)x5−7(−2)x−3+2(−5)x−6f′(x)=−1+36x5+14x−3−10x−6f′(x)=−1+36x5+14x3−10x6\begin{aligned} f(x) &= 9 - x + 6x^6 - \frac{7}{x^2} + \frac{2}{x^5} \\ f(x) &= 9 - x + 6x^6 - 7x^{-2} + 2x^{-5} \\ f'(x) &= -1 + 6(6)x^5 - 7(-2)x^{-3} + 2(-5)x^{-6} \\ f'(x) &= -1 + 36x^5 + 14x^{-3} - 10x^{-6} \\ \mathbf{f'(x)} &= \mathbf{-1 + 36x^5 + \frac{14}{x^3} - \frac{10}{x^6}} \end{aligned}
f(x)=−1x8+5x3+3x+2−4xf(x)=−x−8+5x3+3x−1+2−4xf′(x)=−(−8)x−9+5(3)x2+3(−1)x−2−4f′(x)=8x−9+15x2−3x−2−4f′(x)=8x9+15x2−3x2−4\begin{aligned} f(x) &= -\frac{1}{x^8} + 5x^3 + \frac{3}{x} + 2 - 4x \\ f(x) &= -x^{-8} + 5x^3 + 3x^{-1} + 2 - 4x \\ f'(x) &= -(-8)x^{-9} + 5(3)x^2 + 3(-1)x^{-2} - 4 \\ f'(x) &= 8x^{-9} + 15x^2 - 3x^{-2} - 4 \\ \mathbf{f'(x)} &= \mathbf{\frac{8}{x^9} + 15x^2 - \frac{3}{x^2} - 4} \end{aligned}
f(x)=2x7−10+8x3+4x−3x5f(x)=2x7−10+8x−3+4x−3x−5f′(x)=2(7)x6+8(−3)x−4+4−3(−5)x−6f′(x)=14x6−24x−4+4+15x−6f′(x)=14x6−24x4+4+15x6\begin{aligned} f(x) &= 2x^7 - 10 + \frac{8}{x^3} + 4x - \frac{3}{x^5} \\ f(x) &= 2x^7 - 10 + 8x^{-3} + 4x - 3x^{-5} \\ f'(x) &= 2(7)x^6 + 8(-3)x^{-4} + 4 - 3(-5)x^{-6} \\ f'(x) &= 14x^6 - 24x^{-4} + 4 + 15x^{-6} \\ \mathbf{f'(x)} &= \mathbf{14x^6 - \frac{24}{x^4} + 4 + \frac{15}{x^6}} \end{aligned}

Ejercicios extra 1

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Bloque A
f(x)=8x5−4x3+2x2−9f(x) = 8x^5 - 4x^3 + 2x^2 - 9
f(x)=−3x4+5x2−7x+10f(x) = -3x^4 + 5x^2 - 7x + 10
f(x)=10x3−7x2+6x−12f(x) = 10x^3 - 7x^2 + 6x - 12
f(x)=−x9+4x5−2x3+x−5f(x) = -x^9 + 4x^5 - 2x^3 + x - 5
f(x)=5x4−2x3+7x2−4x+1f(x) = 5x^4 - 2x^3 + 7x^2 - 4x + 1
f(x)=−6x7+3x4−8x2+5x−2f(x) = -6x^7 + 3x^4 - 8x^2 + 5x - 2
Bloque B
f(x)=3x−6x4+1x10−2+4x2f(x) = 3x - 6x^4 + \dfrac{1}{x^{10}} - 2 + \dfrac{4}{x^2}
f(x)=12+2x7−6x7−3x+5x3f(x) = 12 + 2x^7 - \dfrac{6}{x^7} - 3x + \dfrac{5}{x^3}
f(x)=−6x2−2x+9x2+3−2xf(x) = -6x^2 - 2x + \dfrac{9}{x^2} + 3 - \dfrac{2}{x}
f(x)=2x4−11−4x3+2x−3x6f(x) = \dfrac{2}{x^4} - 11 - 4x^3 + 2x - \dfrac{3}{x^6}
f(x)=7x5+6x−9+2x−4x3f(x) = 7x^5 + \dfrac{6}{x} - 9 + 2x - \dfrac{4}{x^3}
f(x)=−3x+6x3+8−5x4−2x5f(x) = -3x + \dfrac{6}{x^3} + 8 - 5x^4 - \dfrac{2}{x^5}

Ejercicios extra 2

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Bloque C
f(x)=7x3−2x6+4xf(x) = \dfrac{7}{x^3} - \dfrac{2}{x^6} + \dfrac{4}{x}
f(x)=−5x7+3x2−1x5f(x) = -\dfrac{5}{x^7} + \dfrac{3}{x^2} - \dfrac{1}{x^5}
f(x)=4x4−3x5+9x2f(x) = \dfrac{4}{x^4} - \dfrac{3}{x^5} + \dfrac{9}{x^2}
f(x)=−8x3+5x6−2x2f(x) = -\dfrac{8}{x^3} + \dfrac{5}{x^6} - \dfrac{2}{x^2}
f(x)=5x2+3x4−2x7f(x) = \dfrac{5}{x^2} + \dfrac{3}{x^4} - \dfrac{2}{x^7}
f(x)=−6x4−3x5+7x2f(x) = -\dfrac{6}{x^4} - \dfrac{3}{x^5} + \dfrac{7}{x^2}
f(x)=−1x6+4x3−2x5f(x) = -\dfrac{1}{x^6} + \dfrac{4}{x^3} - \dfrac{2}{x^5}
f(x)=−2x8−3x5+8x4f(x) = -\dfrac{2}{x^8} - \dfrac{3}{x^5} + \dfrac{8}{x^4}
Bloque D
f(x)=9x5−4x4+2x2−6x+1f(x) = 9x^5 - 4x^4 + 2x^2 - 6x + 1
f(x)=−3x6+7x3−5x+4f(x) = -3x^6 + 7x^3 - 5x + 4
f(x)=4x8−2x5+3x3−x−7f(x) = 4x^8 - 2x^5 + 3x^3 - x - 7
f(x)=−x4+6x3−8x2+2f(x) = -x^4 + 6x^3 - 8x^2 + 2
Bloque E
f(x)=−1+2x5+3x−7x4+3x2f(x) = -1 + 2x^5 + 3x - \dfrac{7}{x^4} + \dfrac{3}{x^2}
f(x)=4x3−3x2+5−6x+5x6f(x) = 4x^3 - \dfrac{3}{x^2} + 5 - 6x + \dfrac{5}{x^6}
f(x)=1x7−2x+8x4−4x−9f(x) = \dfrac{1}{x^7} - 2x + 8x^4 - \dfrac{4}{x} - 9
f(x)=3x+2−4x6+9x4−5xf(x) = 3x + 2 - 4x^6 + \dfrac{9}{x^4} - \dfrac{5}{x}
f(x)=8x−5+4x4−2x4−6xf(x) = \dfrac{8}{x} - 5 + 4x^4 - \dfrac{2}{x^4} - 6x
f(x)=3x5−7x+1−2x6+5x3f(x) = 3x^5 - 7x + 1 - \dfrac{2}{x^6} + \dfrac{5}{x^3}
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