Inicio › Matemáticas 5 › Derivadas de raíces

Matemáticas 5 · Tema 4

Derivadas de raíces

Apuntes

Para derivar una raíz, primero la transformamos en un exponente fraccionario usando la regla:

xmn=xmn\sqrt[n]{x^m} = x^{\frac{m}{n}}

Ejemplos

f(x)=−6x43f(x)=−6x43f′(x)=−6(43)x43−33=(−61)(43)x13=−243x13=−8x3\begin{aligned} f(x) &= -6\sqrt[3]{x^4} \\ f(x) &= -6x^{\frac{4}{3}} \\ f'(x) &= -6\left(\frac{4}{3}\right)x^{\frac{4}{3} - \frac{3}{3}} \\ &= \left(-\frac{6}{1}\right)\left(\frac{4}{3}\right)x^{\frac{1}{3}} \\ &= -\frac{24}{3}x^{\frac{1}{3}} \\ &= \mathbf{-8\sqrt[3]{x}} \end{aligned}
g(x)=5x4g(x)=5x14g′(x)=(5)(14)x14−44=51(14)x−34=54x−34=54x34=54x34\begin{aligned} g(x) &= 5\sqrt[4]{x} \\ g(x) &= 5x^{\frac{1}{4}} \\ g'(x) &= (5)\left(\frac{1}{4}\right)x^{\frac{1}{4} - \frac{4}{4}} \\ &= \frac{5}{1}\left(\frac{1}{4}\right)x^{-\frac{3}{4}} \\ &= \frac{5}{4}x^{-\frac{3}{4}} \\ &= \frac{5}{4x^{\frac{3}{4}}} \\ &= \mathbf{\frac{5}{4\sqrt[4]{x^3}}} \end{aligned}
h(x)=−56x23h(x)=(−56)x−23h′(x)=(−56)(−23)x−23−33=1018x−53=1018x53=59x53\begin{aligned} h(x) &= \frac{-5}{6\sqrt[3]{x^2}} \\ h(x) &= \left(-\frac{5}{6}\right)x^{-\frac{2}{3}} \\ h'(x) &= \left(-\frac{5}{6}\right)\left(-\frac{2}{3}\right)x^{-\frac{2}{3} - \frac{3}{3}} \\ &= \frac{10}{18}x^{-\frac{5}{3}} \\ &= \frac{10}{18x^{\frac{5}{3}}} \\ &= \mathbf{\frac{5}{9\sqrt[3]{x^5}}} \end{aligned}

Ejercicios

Bloque 1
f(x)=−4x5f(x) = -4\sqrt{x^5}
f(x)=−5x3f(x) = -5\sqrt{x^3}
f(x)=−4x54f(x) = -4\sqrt[4]{x^5}
f(x)=3x53f(x) = 3\sqrt[3]{x^5}
f(x)=−2x75f(x) = -2\sqrt[5]{x^7}
f(x)=8x74f(x) = 8\sqrt[4]{x^7}
Bloque 2
f(x)=7x4f(x) = 7\sqrt[4]{x}
f(x)=−9x25f(x) = -9\sqrt[5]{x^2}
f(x)=8x23f(x) = 8\sqrt[3]{x^2}
f(x)=−2x35f(x) = -2\sqrt[5]{x^3}
f(x)=6xf(x) = 6\sqrt{x}
f(x)=−6x34f(x) = -6\sqrt[4]{x^3}
Bloque 3
f(x)=−26x43f(x) = \dfrac{-2}{6\sqrt[3]{x^4}}
f(x)=−64x23f(x) = \dfrac{-6}{4\sqrt[3]{x^2}}
f(x)=−39x3f(x) = \dfrac{-3}{9\sqrt{x^3}}
f(x)=−105x4f(x) = \dfrac{-10}{5\sqrt[4]{x}}
f(x)=−46x35f(x) = \dfrac{-4}{6\sqrt[5]{x^3}}
f(x)=812xf(x) = \dfrac{8}{12\sqrt{x}}

Respuestas

▼
Bloque 1
f(x)=−4x5f(x)=−4x52f′(x)=−4(52)x52−22=−202x32=−10x3\begin{aligned} f(x) &= -4\sqrt{x^5} \\ f(x) &= -4x^{\frac{5}{2}} \\ f'(x) &= -4\left(\frac{5}{2}\right)x^{\frac{5}{2} - \frac{2}{2}} \\ &= -\frac{20}{2}x^{\frac{3}{2}} \\ &= \mathbf{-10\sqrt{x^3}} \end{aligned}
f(x)=−5x3f(x)=−5x32f′(x)=−5(32)x32−22=−152x\begin{aligned} f(x) &= -5\sqrt{x^3} \\ f(x) &= -5x^{\frac{3}{2}} \\ f'(x) &= -5\left(\frac{3}{2}\right)x^{\frac{3}{2} - \frac{2}{2}} \\ &= \mathbf{-\frac{15}{2}\sqrt{x}} \end{aligned}
f(x)=−4x54f(x)=−4x54f′(x)=−4(54)x54−44=−5x14=−5x4\begin{aligned} f(x) &= -4\sqrt[4]{x^5} \\ f(x) &= -4x^{\frac{5}{4}} \\ f'(x) &= -4\left(\frac{5}{4}\right)x^{\frac{5}{4} - \frac{4}{4}} \\ &= -5x^{\frac{1}{4}} \\ &= \mathbf{-5\sqrt[4]{x}} \end{aligned}
f(x)=3x53f(x)=3x53f′(x)=3(53)x53−33=5x23=5x23\begin{aligned} f(x) &= 3\sqrt[3]{x^5} \\ f(x) &= 3x^{\frac{5}{3}} \\ f'(x) &= 3\left(\frac{5}{3}\right)x^{\frac{5}{3} - \frac{3}{3}} \\ &= 5x^{\frac{2}{3}} \\ &= \mathbf{5\sqrt[3]{x^2}} \end{aligned}
f(x)=−2x75f(x)=−2x75f′(x)=−2(75)x75−55=−145x25=−145x25\begin{aligned} f(x) &= -2\sqrt[5]{x^7} \\ f(x) &= -2x^{\frac{7}{5}} \\ f'(x) &= -2\left(\frac{7}{5}\right)x^{\frac{7}{5} - \frac{5}{5}} \\ &= -\frac{14}{5}x^{\frac{2}{5}} \\ &= \mathbf{-\frac{14}{5}\sqrt[5]{x^2}} \end{aligned}
f(x)=8x74f(x)=8x74f′(x)=8(74)x74−44=14x34=14x34\begin{aligned} f(x) &= 8\sqrt[4]{x^7} \\ f(x) &= 8x^{\frac{7}{4}} \\ f'(x) &= 8\left(\frac{7}{4}\right)x^{\frac{7}{4} - \frac{4}{4}} \\ &= 14x^{\frac{3}{4}} \\ &= \mathbf{14\sqrt[4]{x^3}} \end{aligned}
Bloque 2
f(x)=7x4f(x)=7x14f′(x)=7(14)x14−44=74x−34=74x34\begin{aligned} f(x) &= 7\sqrt[4]{x} \\ f(x) &= 7x^{\frac{1}{4}} \\ f'(x) &= 7\left(\frac{1}{4}\right)x^{\frac{1}{4} - \frac{4}{4}} \\ &= \frac{7}{4}x^{-\frac{3}{4}} \\ &= \mathbf{\frac{7}{4\sqrt[4]{x^3}}} \end{aligned}
f(x)=−9x25f(x)=−9x25f′(x)=−9(25)x25−55=−185x−35=−185x35\begin{aligned} f(x) &= -9\sqrt[5]{x^2} \\ f(x) &= -9x^{\frac{2}{5}} \\ f'(x) &= -9\left(\frac{2}{5}\right)x^{\frac{2}{5} - \frac{5}{5}} \\ &= -\frac{18}{5}x^{-\frac{3}{5}} \\ &= \mathbf{-\frac{18}{5\sqrt[5]{x^3}}} \end{aligned}
f(x)=8x23f(x)=8x23f′(x)=8(23)x23−33=163x−13=163x3\begin{aligned} f(x) &= 8\sqrt[3]{x^2} \\ f(x) &= 8x^{\frac{2}{3}} \\ f'(x) &= 8\left(\frac{2}{3}\right)x^{\frac{2}{3} - \frac{3}{3}} \\ &= \frac{16}{3}x^{-\frac{1}{3}} \\ &= \mathbf{\frac{16}{3\sqrt[3]{x}}} \end{aligned}
f(x)=−2x35f(x)=−2x35f′(x)=−2(35)x35−55=−65x−25=−65x25\begin{aligned} f(x) &= -2\sqrt[5]{x^3} \\ f(x) &= -2x^{\frac{3}{5}} \\ f'(x) &= -2\left(\frac{3}{5}\right)x^{\frac{3}{5} - \frac{5}{5}} \\ &= -\frac{6}{5}x^{-\frac{2}{5}} \\ &= \mathbf{-\frac{6}{5\sqrt[5]{x^2}}} \end{aligned}
f(x)=6xf(x)=6x12f′(x)=6(12)x12−22=3x−12=3x\begin{aligned} f(x) &= 6\sqrt{x} \\ f(x) &= 6x^{\frac{1}{2}} \\ f'(x) &= 6\left(\frac{1}{2}\right)x^{\frac{1}{2} - \frac{2}{2}} \\ &= 3x^{-\frac{1}{2}} \\ &= \mathbf{\frac{3}{\sqrt{x}}} \end{aligned}
f(x)=−6x34f(x)=−6x34f′(x)=−6(34)x34−44=−92x−14=−92x4\begin{aligned} f(x) &= -6\sqrt[4]{x^3} \\ f(x) &= -6x^{\frac{3}{4}} \\ f'(x) &= -6\left(\frac{3}{4}\right)x^{\frac{3}{4} - \frac{4}{4}} \\ &= -\frac{9}{2}x^{-\frac{1}{4}} \\ &= \mathbf{-\frac{9}{2\sqrt[4]{x}}} \end{aligned}
Bloque 3
f(x)=−26x43f(x)=−13x−43f′(x)=(−13)(−43)x−43−33=49x−73=49x73\begin{aligned} f(x) &= \frac{-2}{6\sqrt[3]{x^4}} \\ f(x) &= -\frac{1}{3}x^{-\frac{4}{3}} \\ f'(x) &= \left(-\frac{1}{3}\right)\left(-\frac{4}{3}\right)x^{-\frac{4}{3} - \frac{3}{3}} \\ &= \frac{4}{9}x^{-\frac{7}{3}} \\ &= \mathbf{\frac{4}{9\sqrt[3]{x^7}}} \end{aligned}
f(x)=−64x23f(x)=−32x−23f′(x)=(−32)(−23)x−23−33=1x−53=1x53\begin{aligned} f(x) &= \frac{-6}{4\sqrt[3]{x^2}} \\ f(x) &= -\frac{3}{2}x^{-\frac{2}{3}} \\ f'(x) &= \left(-\frac{3}{2}\right)\left(-\frac{2}{3}\right)x^{-\frac{2}{3} - \frac{3}{3}} \\ &= 1x^{-\frac{5}{3}} \\ &= \mathbf{\frac{1}{\sqrt[3]{x^5}}} \end{aligned}
f(x)=−39x3f(x)=−13x−32f′(x)=(−13)(−32)x−32−22=12x−52=12x5\begin{aligned} f(x) &= \frac{-3}{9\sqrt{x^3}} \\ f(x) &= -\frac{1}{3}x^{-\frac{3}{2}} \\ f'(x) &= \left(-\frac{1}{3}\right)\left(-\frac{3}{2}\right)x^{-\frac{3}{2} - \frac{2}{2}} \\ &= \frac{1}{2}x^{-\frac{5}{2}} \\ &= \mathbf{\frac{1}{2\sqrt{x^5}}} \end{aligned}
f(x)=−105x4f(x)=−2x−14f′(x)=(−2)(−14)x−14−44=12x−54=12x54\begin{aligned} f(x) &= \frac{-10}{5\sqrt[4]{x}} \\ f(x) &= -2x^{-\frac{1}{4}} \\ f'(x) &= (-2)\left(-\frac{1}{4}\right)x^{-\frac{1}{4} - \frac{4}{4}} \\ &= \frac{1}{2}x^{-\frac{5}{4}} \\ &= \mathbf{\frac{1}{2\sqrt[4]{x^5}}} \end{aligned}
f(x)=−46x35f(x)=−23x−35f′(x)=(−23)(−35)x−35−55=25x−85=25x85\begin{aligned} f(x) &= \frac{-4}{6\sqrt[5]{x^3}} \\ f(x) &= -\frac{2}{3}x^{-\frac{3}{5}} \\ f'(x) &= \left(-\frac{2}{3}\right)\left(-\frac{3}{5}\right)x^{-\frac{3}{5} - \frac{5}{5}} \\ &= \frac{2}{5}x^{-\frac{8}{5}} \\ &= \mathbf{\frac{2}{5\sqrt[5]{x^8}}} \end{aligned}
f(x)=812xf(x)=23x−12f′(x)=(23)(−12)x−12−22=−13x−32=−13x3\begin{aligned} f(x) &= \frac{8}{12\sqrt{x}} \\ f(x) &= \frac{2}{3}x^{-\frac{1}{2}} \\ f'(x) &= \left(\frac{2}{3}\right)\left(-\frac{1}{2}\right)x^{-\frac{1}{2} - \frac{2}{2}} \\ &= -\frac{1}{3}x^{-\frac{3}{2}} \\ &= \mathbf{-\frac{1}{3\sqrt{x^3}}} \end{aligned}

Ejercicios extra 1

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Bloque A
f(x)=10x7f(x) = 10\sqrt{x^7}
f(x)=−6x94f(x) = -6\sqrt[4]{x^9}
f(x)=−7x3f(x) = -7\sqrt{x^3}
f(x)=−9x85f(x) = -9\sqrt[5]{x^8}
f(x)=−8x73f(x) = -8\sqrt[3]{x^7}
f(x)=−3x76f(x) = -3\sqrt[6]{x^7}
Bloque B
f(x)=−3x45f(x) = -3\sqrt[5]{x^4}
f(x)=12x56f(x) = 12\sqrt[6]{x^5}
f(x)=4x3f(x) = 4\sqrt[3]{x}
f(x)=2x23f(x) = 2\sqrt[3]{x^2}
f(x)=−5x6f(x) = -5\sqrt[6]{x}
f(x)=10x5f(x) = 10\sqrt[5]{x}
Bloque C
f(x)=−15x23f(x) = \dfrac{-1}{5\sqrt[3]{x^2}}
f(x)=−82x3f(x) = \dfrac{-8}{2\sqrt{x^3}}
f(x)=510x34f(x) = \dfrac{5}{10\sqrt[4]{x^3}}
f(x)=−36x53f(x) = \dfrac{-3}{6\sqrt[3]{x^5}}
f(x)=−48x25f(x) = \dfrac{-4}{8\sqrt[5]{x^2}}
f(x)=69x5f(x) = \dfrac{6}{9\sqrt{x^5}}

Ejercicios extra 2

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Bloque D
f(x)=5x83f(x) = 5\sqrt[3]{x^8}
f(x)=−2x9f(x) = -2\sqrt{x^9}
f(x)=6x114f(x) = 6\sqrt[4]{x^{11}}
f(x)=−4x95f(x) = -4\sqrt[5]{x^9}
Bloque E
f(x)=−10x3f(x) = -10\sqrt[3]{x}
f(x)=15x25f(x) = 15\sqrt[5]{x^2}
f(x)=−8x4f(x) = -8\sqrt[4]{x}
f(x)=12x34f(x) = 12\sqrt[4]{x^3}
Bloque F
f(x)=34x3f(x) = \dfrac{3}{4\sqrt[3]{x}}
f(x)=−96x54f(x) = \dfrac{-9}{6\sqrt[4]{x^5}}
f(x)=28x6f(x) = \dfrac{2}{8\sqrt[6]{x}}
f(x)=−515x7f(x) = \dfrac{-5}{15\sqrt{x^7}}
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