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Find the derivative of $y=\ln\left(\sqrt{\arctan\left(x\right)}\right)^{-1}$

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Solution

Final answer to the problem

$\frac{-\frac{1}{2}}{\left(1+x^2\right)\ln\left(\sqrt{\arctan\left(x\right)}\right)^{2}\arctan\left(x\right)}$
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Step-by-step Solution

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The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$-\ln\left(\sqrt{\arctan\left(x\right)}\right)^{-2}\frac{d}{dx}\left(\ln\left(\sqrt{\arctan\left(x\right)}\right)\right)$

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$-\ln\left(\sqrt{\arctan\left(x\right)}\right)^{-2}\frac{d}{dx}\left(\ln\left(\sqrt{\arctan\left(x\right)}\right)\right)$

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Learn how to solve problems step by step online. Find the derivative of y=ln(arctan(x)^1/2)^(-1). 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 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}. Multiplying the fraction by -1. The power rule for differentiation states that if n is a real number and f(x) = x^n, then f'(x) = nx^{n-1}.

Final answer to the problem

$\frac{-\frac{1}{2}}{\left(1+x^2\right)\ln\left(\sqrt{\arctan\left(x\right)}\right)^{2}\arctan\left(x\right)}$

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

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Function Plot

Plotting: $\frac{-\frac{1}{2}}{\left(1+x^2\right)\ln\left(\sqrt{\arctan\left(x\right)}\right)^{2}\arctan\left(x\right)}$

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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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