Inicio › Matemáticas 5 › Regla del producto (P2)

Matemáticas 5 · Tema 6

Regla del producto (P2)

Apuntes

La misma regla del producto, ahora con trigonométricas, logaritmos, exponenciales y raíces:

f(x)=u⋅v  ⟹  f′(x)=u′v+uv′f(x) = u \cdot v \implies f'(x) = u'v + uv'

Ejemplos

f(x)=5x3cos⁡xf′(x)=(15x2)(cos⁡x)+(5x3)(−sen⁡x)f′(x)=15x2cos⁡x−5x3sen⁡x\begin{aligned} f(x) &= 5x^3 \cos x \\ f'(x) &= (15x^2)(\cos x) + (5x^3)(-\operatorname{sen} x) \\ \mathbf{f'(x)} &= \mathbf{15x^2 \cos x - 5x^3 \operatorname{sen} x} \end{aligned}
f(x)=7sen⁡xcos⁡xf′(x)=(7cos⁡x)(cos⁡x)+(7sen⁡x)(−sen⁡x)f′(x)=7cos⁡2x−7sen⁡2x\begin{aligned} f(x) &= 7 \operatorname{sen} x \cos x \\ f'(x) &= (7 \cos x)(\cos x) + (7 \operatorname{sen} x)(-\operatorname{sen} x) \\ \mathbf{f'(x)} &= \mathbf{7 \cos^2 x - 7 \operatorname{sen}^2 x} \end{aligned}
f(x)=x53ln⁡x=x53ln⁡xf′(x)=(53x53−33)(ln⁡x)+(x53)(1x)f′(x)=53x23ln⁡x+x53xf′(x)=53x23ln⁡x+x53x\begin{aligned} f(x) &= \sqrt[3]{x^5} \ln x = x^{\frac{5}{3}} \ln x \\ f'(x) &= \left( \frac{5}{3} x^{\frac{5}{3} - \frac{3}{3}} \right)(\ln x) + (x^{\frac{5}{3}})\left( \frac{1}{x} \right) \\ f'(x) &= \frac{5}{3} x^{\frac{2}{3}} \ln x + \frac{x^{\frac{5}{3}}}{x} \\ \mathbf{f'(x)} &= \mathbf{\frac{5}{3} \sqrt[3]{x^2} \ln x + \frac{\sqrt[3]{x^5}}{x}} \end{aligned}
f(x)=x5 7x=x157xf′(x)=(15x15−55)(7x)+(x15)(7xln⁡7)f′(x)=15x−457x+x5 7xln⁡7f′(x)=15x457x+x5 7xln⁡7\begin{aligned} f(x) &= \sqrt[5]{x} \, 7^x = x^{\frac{1}{5}} 7^x \\ f'(x) &= \left( \frac{1}{5} x^{\frac{1}{5} - \frac{5}{5}} \right)(7^x) + (x^{\frac{1}{5}})(7^x \ln 7) \\ f'(x) &= \frac{1}{5} x^{-\frac{4}{5}} 7^x + \sqrt[5]{x} \, 7^x \ln 7 \\ \mathbf{f'(x)} &= \mathbf{\frac{1}{5 \sqrt[5]{x^4}} 7^x + \sqrt[5]{x} \, 7^x \ln 7} \end{aligned}

Ejercicios

Bloque 1
f(x)=−4x2sen⁡xf(x) = -4x^2 \operatorname{sen} x
f(x)=5x4cos⁡xf(x) = 5x^4 \cos x
Bloque 2
f(x)=8sen⁡xtan⁡xf(x) = 8 \operatorname{sen} x \tan x
f(x)=−4sen⁡xcos⁡xf(x) = -4 \operatorname{sen} x \cos x
Bloque 3
f(x)=xln⁡xf(x) = \sqrt{x} \ln x
f(x)=x53 6xf(x) = \sqrt[3]{x^5} \, 6^x

Respuestas

▼
Bloque 1
f(x)=−4x2sen⁡xf′(x)=(−8x)(sen⁡x)+(−4x2)(cos⁡x)f′(x)=−8xsen⁡x−4x2cos⁡x\begin{aligned} f(x) &= -4x^2 \operatorname{sen} x \\ f'(x) &= (-8x)(\operatorname{sen} x) + (-4x^2)(\cos x) \\ \mathbf{f'(x)} &= \mathbf{-8x \operatorname{sen} x - 4x^2 \cos x} \end{aligned}
f(x)=5x4cos⁡xf′(x)=(20x3)(cos⁡x)+(5x4)(−sen⁡x)f′(x)=20x3cos⁡x−5x4sen⁡x\begin{aligned} f(x) &= 5x^4 \cos x \\ f'(x) &= (20x^3)(\cos x) + (5x^4)(-\operatorname{sen} x) \\ \mathbf{f'(x)} &= \mathbf{20x^3 \cos x - 5x^4 \operatorname{sen} x} \end{aligned}
Bloque 2
f(x)=8sen⁡xtan⁡xf′(x)=(8cos⁡x)(tan⁡x)+(8sen⁡x)(sec⁡2x)f′(x)=8cos⁡xtan⁡x+8sen⁡xsec⁡2x\begin{aligned} f(x) &= 8 \operatorname{sen} x \tan x \\ f'(x) &= (8 \cos x)(\tan x) + (8 \operatorname{sen} x)(\sec^2 x) \\ \mathbf{f'(x)} &= \mathbf{8 \cos x \tan x + 8 \operatorname{sen} x \sec^2 x} \end{aligned}
f(x)=−4sen⁡xcos⁡xf′(x)=(−4cos⁡x)(cos⁡x)+(−4sen⁡x)(−sen⁡x)f′(x)=−4cos⁡2x+4sen⁡2x\begin{aligned} f(x) &= -4 \operatorname{sen} x \cos x \\ f'(x) &= (-4 \cos x)(\cos x) + (-4 \operatorname{sen} x)(-\operatorname{sen} x) \\ \mathbf{f'(x)} &= \mathbf{-4 \cos^2 x + 4 \operatorname{sen}^2 x} \end{aligned}
Bloque 3
f(x)=xln⁡x=x12ln⁡xf′(x)=(12x−12)(ln⁡x)+(x12)(1x)f′(x)=ln⁡x2x+1x\begin{aligned} f(x) &= \sqrt{x} \ln x = x^{\frac{1}{2}} \ln x \\ f'(x) &= \left( \frac{1}{2} x^{-\frac{1}{2}} \right)(\ln x) + (x^{\frac{1}{2}})\left( \frac{1}{x} \right) \\ \mathbf{f'(x)} &= \mathbf{\frac{\ln x}{2\sqrt{x}} + \frac{1}{\sqrt{x}}} \end{aligned}
f(x)=x53 6x=x536xf′(x)=(53x23)6x+(x53)(6xln⁡6)f′(x)=53x23 6x+x53 6xln⁡6\begin{aligned} f(x) &= \sqrt[3]{x^5} \, 6^x = x^{\frac{5}{3}} 6^x \\ f'(x) &= \left( \frac{5}{3} x^{\frac{2}{3}} \right)6^x + (x^{\frac{5}{3}})(6^x \ln 6) \\ \mathbf{f'(x)} &= \mathbf{\frac{5}{3} \sqrt[3]{x^2} \, 6^x + \sqrt[3]{x^5} \, 6^x \ln 6} \end{aligned}

Ejercicios extra 1

▼
Bloque A
f(x)=−3x2cos⁡xf(x) = -3x^2 \cos x
f(x)=4x3tan⁡xf(x) = 4x^3 \tan x
f(x)=−2x5ln⁡xf(x) = -2x^5 \ln x
f(x)=7x2exf(x) = 7x^2 e^x
Bloque B
f(x)=5sen⁡xln⁡xf(x) = 5 \operatorname{sen} x \ln x
f(x)=2xcos⁡xf(x) = 2^x \cos x
f(x)=4xsen⁡xf(x) = 4^x \operatorname{sen} x
f(x)=cos⁡xln⁡xf(x) = \cos x \ln x
Bloque C
f(x)=x exf(x) = \sqrt{x} \, e^x
f(x)=x23sen⁡xf(x) = \sqrt[3]{x^2} \operatorname{sen} x
f(x)=x35tan⁡xf(x) = \sqrt[5]{x^3} \tan x
f(x)=xsen⁡xf(x) = \sqrt{x} \operatorname{sen} x

Ejercicios extra 2

▼
Bloque D
f(x)=3x4ln⁡xf(x) = 3x^4 \ln x
f(x)=−5x3exf(x) = -5x^3 e^x
f(x)=8x2 3xf(x) = 8x^2 \, 3^x
f(x)=−6x5cos⁡xf(x) = -6x^5 \cos x
f(x)=6x2tan⁡xf(x) = 6x^2 \tan x
Bloque E
f(x)=4tan⁡xln⁡xf(x) = 4 \tan x \ln x
f(x)=−2excos⁡xf(x) = -2e^x \cos x
f(x)=7exsen⁡xf(x) = 7e^x \operatorname{sen} x
f(x)=5xtan⁡xf(x) = 5^x \tan x
f(x)=6cos⁡xsen⁡xf(x) = 6 \cos x \operatorname{sen} x
Bloque F
f(x)=x3cos⁡xf(x) = \sqrt[3]{x} \cos x
f(x)=x34ln⁡xf(x) = \sqrt[4]{x^3} \ln x
f(x)=x3 exf(x) = \sqrt{x^3} \, e^x
f(x)=x25 3xf(x) = \sqrt[5]{x^2} \, 3^x
f(x)=x43ln⁡xf(x) = \sqrt[3]{x^4} \ln x
f(x)=x43exf(x) = \sqrt[3]{x^4} e^x
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