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Find the derivative of $\arccos\left(\frac{x^2+2}{x^2-2}\right)$

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Solution

Final Answer

$\frac{-\left(2x\left(x^2-2\right)+2\left(-x^2-2\right)x\right)}{\sqrt{1+\frac{-x^{4}-4x^2-4}{\left(x^2-2\right)^2}}\left(x^2-2\right)^2}$
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Step-by-step Solution

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Taking the derivative of arccosine

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

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

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Learn how to solve problems step by step online. Find the derivative of arccos((x^2+2)/(x^2-2)). Taking the derivative of arccosine. The power of a quotient is equal to the quotient of the power of the numerator and denominator: \displaystyle\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}. Multiplying the fraction by -1. Apply the quotient rule for differentiation, which states that if f(x) and g(x) are functions and h(x) is the function defined by {\displaystyle h(x) = \frac{f(x)}{g(x)}}, where {g(x) \neq 0}, then {\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}.

Final Answer

$\frac{-\left(2x\left(x^2-2\right)+2\left(-x^2-2\right)x\right)}{\sqrt{1+\frac{-x^{4}-4x^2-4}{\left(x^2-2\right)^2}}\left(x^2-2\right)^2}$

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 arccos((x^2+2)/(x^2-2)) using the product ruleFind derivative of arccos((x^2+2)/(x^2-2)) using the quotient ruleFind derivative of arccos((x^2+2)/(x^2-2)) using logarithmic differentiationFind derivative of arccos((x^2+2)/(x^2-2)) using the definition

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Function Plot

Plotting: $\frac{-\left(2x\left(x^2-2\right)+2\left(-x^2-2\right)x\right)}{\sqrt{1+\frac{-x^{4}-4x^2-4}{\left(x^2-2\right)^2}}\left(x^2-2\right)^2}$

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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
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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