Find the derivative of $\ln\left(x^{8\cos\left(x\right)}\right)$

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Final answer to the problem

$-8\sin\left(x\right)\ln\left(x\right)+\frac{8\cos\left(x\right)}{x}$
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Step-by-step Solution

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  • Find the derivative using the definition
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  • Find the derivative using logarithmic differentiation
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The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If $f(x)=ln\:a$ (where $a$ is a function of $x$), then $\displaystyle f'(x)=\frac{a'}{a}$

Learn how to solve one-variable linear inequalities problems step by step online.

$\frac{1}{x^{8\cos\left(x\right)}}\frac{d}{dx}\left(x^{8\cos\left(x\right)}\right)$

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Learn how to solve one-variable linear inequalities problems step by step online. Find the derivative of ln(x^(8cos(x))). The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If f(x)=ln\:a (where a is a function of x), then \displaystyle f'(x)=\frac{a'}{a}. The derivative \frac{d}{dx}\left(x^{8\cos\left(x\right)}\right) results in 8\left(-\sin\left(x\right)\ln\left(x\right)+\frac{\cos\left(x\right)}{x}\right)x^{8\cos\left(x\right)}. Multiplying the fraction by 8\left(-\sin\left(x\right)\ln\left(x\right)+\frac{\cos\left(x\right)}{x}\right)x^{8\cos\left(x\right)}. Any expression multiplied by 1 is equal to itself.

Final answer to the problem

$-8\sin\left(x\right)\ln\left(x\right)+\frac{8\cos\left(x\right)}{x}$

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Plotting: $-8\sin\left(x\right)\ln\left(x\right)+\frac{8\cos\left(x\right)}{x}$

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7
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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
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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