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Find the derivative using logarithmic differentiation method $\frac{d}{dx}\left(\ln\left(x+9\right)^x\right)$

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Solution

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Final answer to the problem

$\left(\ln\left(\ln\left(x+9\right)\right)+\frac{x}{\left(x+9\right)\ln\left(x+9\right)}\right)\ln\left(x+9\right)^x$
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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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To derive the function $\ln\left(x+9\right)^x$, use the method of logarithmic differentiation. First, assign the function to $y$, then take the natural logarithm of both sides of the equation

$y=\ln\left(x+9\right)^x$

Learn how to solve logarithmic differentiation problems step by step online.

$y=\ln\left(x+9\right)^x$

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Learn how to solve logarithmic differentiation problems step by step online. Find the derivative using logarithmic differentiation method d/dx(ln(x+9)^x). To derive the function \ln\left(x+9\right)^x, use the method of logarithmic differentiation. First, assign the function to y, then take the natural logarithm of both sides of the equation. Apply natural logarithm to both sides of the equality. Using the power rule of logarithms: \log_a(x^n)=n\cdot\log_a(x). Derive both sides of the equality with respect to x.

Final answer to the problem

$\left(\ln\left(\ln\left(x+9\right)\right)+\frac{x}{\left(x+9\right)\ln\left(x+9\right)}\right)\ln\left(x+9\right)^x$

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Function Plot

Plotting: $\left(\ln\left(\ln\left(x+9\right)\right)+\frac{x}{\left(x+9\right)\ln\left(x+9\right)}\right)\ln\left(x+9\right)^x$

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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
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

How to improve your answer:

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Main Topic: Logarithmic Differentiation

The logarithmic derivative of a function f(x) is defined by the formula f'(x)/f(x).

Used Formulas

See formulas (5)

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