Find the derivative $\frac{d}{dx}\left(\frac{\sqrt[3]{a+bx}}{x}\right)$

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Final answer to the problem

$\frac{-2bx-3a}{3\sqrt[3]{\left(a+bx\right)^{2}}x^2}$
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Step-by-step Solution

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  • Find the derivative using the definition
  • Find the derivative using the product rule
  • Find the derivative using the quotient rule
  • Find the derivative using logarithmic differentiation
  • Find the derivative
  • Integrate by partial fractions
  • Product of Binomials with Common Term
  • FOIL Method
  • Integrate by substitution
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Apply the quotient rule for differentiation, which states that if $f(x)$ and $g(x)$ are functions and $h(x)$ is the function defined by ${\displaystyle h(x) = \frac{f(x)}{g(x)}}$, where ${g(x) \neq 0}$, then ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$

$\frac{\frac{d}{dx}\left(\sqrt[3]{a+bx}\right)x-\sqrt[3]{a+bx}\frac{d}{dx}\left(x\right)}{x^2}$

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$\frac{\frac{d}{dx}\left(\sqrt[3]{a+bx}\right)x-\sqrt[3]{a+bx}\frac{d}{dx}\left(x\right)}{x^2}$

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Learn how to solve problems step by step online. Find the derivative d/dx(((a+bx)^(1/3))/x). Apply the quotient rule for differentiation, which states that if f(x) and g(x) are functions and h(x) is the function defined by {\displaystyle h(x) = \frac{f(x)}{g(x)}}, where {g(x) \neq 0}, then {\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}. The derivative of the linear function is equal to 1. The power rule for differentiation states that if n is a real number and f(x) = x^n, then f'(x) = nx^{n-1}. The derivative of a sum of two or more functions is the sum of the derivatives of each function.

Final answer to the problem

$\frac{-2bx-3a}{3\sqrt[3]{\left(a+bx\right)^{2}}x^2}$

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Function Plot

Plotting: $\frac{-2bx-3a}{3\sqrt[3]{\left(a+bx\right)^{2}}x^2}$

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5
6
7
8
9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
2

e
π
ln
log
log
lim
d/dx
Dx
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

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