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Find the derivative of $\ln\left(\mathrm{tanh}\left(\frac{x}{2}\right)\right)$

Step-by-step Solution

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Final answer to the problem

$\frac{\mathrm{sech}\left(\frac{x}{2}\right)^2}{2\mathrm{tanh}\left(\frac{x}{2}\right)}$
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Step-by-step Solution

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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 $\displaystyle f'(x)=\frac{a'}{a}$

$\frac{1}{\mathrm{tanh}\left(\frac{x}{2}\right)}\frac{d}{dx}\left(\mathrm{tanh}\left(\frac{x}{2}\right)\right)$

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$\frac{1}{\mathrm{tanh}\left(\frac{x}{2}\right)}\frac{d}{dx}\left(\mathrm{tanh}\left(\frac{x}{2}\right)\right)$

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Learn how to solve problems step by step online. Find the derivative of ln(tanh(x/2)). 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}. Taking the derivative of hyperbolic tangent. The derivative of a function multiplied by a constant (\frac{1}{2}) is equal to the constant times the derivative of the function. Divide 1 by 2.

Final answer to the problem

$\frac{\mathrm{sech}\left(\frac{x}{2}\right)^2}{2\mathrm{tanh}\left(\frac{x}{2}\right)}$

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 lntanh(x/2) using the product ruleFind derivative of lntanh(x/2) using the quotient ruleFind derivative of lntanh(x/2) using logarithmic differentiationFind derivative of lntanh(x/2) using the definition

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Function Plot

Plotting: $\frac{\mathrm{sech}\left(\frac{x}{2}\right)^2}{2\mathrm{tanh}\left(\frac{x}{2}\right)}$

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1
2
3
4
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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