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Find the derivative of $\sqrt[5]{4x-5}\ln\left(\frac{2x}{\sqrt[3]{4x-5}}\right)$

Step-by-step Solution

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Solution

Final Answer

$\frac{12x\ln\left(2x\right)-4x\ln\left(4x-5\right)+40x-75}{15\sqrt[5]{\left(4x-5\right)^{4}}x}$
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Step-by-step Solution

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Simplify the derivative by applying the properties of logarithms

$\frac{d}{dx}\left(\sqrt[5]{4x-5}\left(\ln\left(2x\right)-\frac{1}{3}\ln\left(4x-5\right)\right)\right)$

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$\frac{d}{dx}\left(\sqrt[5]{4x-5}\left(\ln\left(2x\right)-\frac{1}{3}\ln\left(4x-5\right)\right)\right)$

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Learn how to solve trigonometric equations problems step by step online. Find the derivative of (4x-5)^1/5ln((2x)/((4x-5)^1/3)). Simplify the derivative by applying the properties of logarithms. Apply the product rule for differentiation: (f\cdot g)'=f'\cdot g+f\cdot g', where f=\sqrt[5]{4x-5} and g=\ln\left(2x\right)-\frac{1}{3}\ln\left(4x-5\right). 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

$\frac{12x\ln\left(2x\right)-4x\ln\left(4x-5\right)+40x-75}{15\sqrt[5]{\left(4x-5\right)^{4}}x}$

Explore different ways to solve this problem

Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more

Find the derivativeFind derivative of (4x-5)^0.2ln(2x/((4x-5)^0.3333)) using the product ruleFind derivative of (4x-5)^0.2ln(2x/((4x-5)^0.3333)) using the quotient ruleFind derivative of (4x-5)^0.2ln(2x/((4x-5)^0.3333)) using logarithmic differentiationFind derivative of (4x-5)^0.2ln(2x/((4x-5)^0.3333)) using the definition

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Function Plot

Plotting: $\frac{12x\ln\left(2x\right)-4x\ln\left(4x-5\right)+40x-75}{15\sqrt[5]{\left(4x-5\right)^{4}}x}$

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

◻√◻
◻√
∞
e
π
ln
log
log◻
lim
d/dx
D□x
∫
∫◻
|◻|
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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