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Find the derivative of $\ln\left(\sqrt{\frac{1+\sin\left(x\right)}{1-\sin\left(x\right)}}\right)$

Step-by-step Solution

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Final Answer

$\sec\left(x\right)$
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Step-by-step Solution

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Simplify the derivative by applying the properties of logarithms

$\frac{d}{dx}\left(\frac{1}{2}\left(\ln\left(1+\sin\left(x\right)\right)-\ln\left(1-\sin\left(x\right)\right)\right)\right)$

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$\frac{d}{dx}\left(\frac{1}{2}\left(\ln\left(1+\sin\left(x\right)\right)-\ln\left(1-\sin\left(x\right)\right)\right)\right)$

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Learn how to solve problems step by step online. Find the derivative of ln(((1+sin(x))/(1-sin(x)))^1/2). Simplify the derivative by applying the properties of logarithms. The derivative of a function multiplied by a constant (\frac{1}{2}) is equal to the constant times the derivative of the function. The derivative of a sum of two or more functions is the sum of the derivatives of each function. The derivative of a function multiplied by a constant (-1) is equal to the constant times the derivative of the function.

Final Answer

$\sec\left(x\right)$

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 ln(((1+sinx)/(1+-1sinx))^0.5) using the product ruleFind derivative of ln(((1+sinx)/(1+-1sinx))^0.5) using the quotient ruleFind derivative of ln(((1+sinx)/(1+-1sinx))^0.5) using logarithmic differentiation

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Function Plot

Plotting: $\sec\left(x\right)$

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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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