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

Step-by-step Solution

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Final Answer

$\frac{\sin\left(\frac{\sqrt{x+1}}{\sqrt{x-1}}\right)}{\sqrt{\left(x-1\right)^{3}}\sqrt{x+1}}$
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Step-by-step Solution

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1

The power of a quotient is equal to the quotient of the power of the numerator and denominator: $\displaystyle\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$

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

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

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Learn how to solve problems step by step online. Find the derivative of cos(((x+1)/(x-1))^1/2). 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}. The derivative of the cosine of a function is equal to minus the sine of the function times the derivative of the function, in other words, if f(x) = \cos(x), then f'(x) = -\sin(x)\cdot D_x(x). 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}}. Cancel exponents \frac{1}{2} and 2.

Final Answer

$\frac{\sin\left(\frac{\sqrt{x+1}}{\sqrt{x-1}}\right)}{\sqrt{\left(x-1\right)^{3}}\sqrt{x+1}}$

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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 cos(((x+1)/(x-1))^0.5) using the product ruleFind derivative of cos(((x+1)/(x-1))^0.5) using the quotient ruleFind derivative of cos(((x+1)/(x-1))^0.5) using logarithmic differentiationFind derivative of cos(((x+1)/(x-1))^0.5) using the definition

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Function Plot

Plotting: $\frac{\sin\left(\frac{\sqrt{x+1}}{\sqrt{x-1}}\right)}{\sqrt{\left(x-1\right)^{3}}\sqrt{x+1}}$

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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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