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Find the derivative of $\frac{x-y}{\left(\frac{1}{2}+\frac{-x}{2y}\right)\left(\sqrt{x-y}-\sqrt{x+y}\right)}$

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Solution

Final Answer

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

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1

Combine all terms into a single fraction with $2y$ as common denominator

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

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

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Learn how to solve problems step by step online. Find the derivative of (x-y)/((1/2+(-x)/(2y))((x-y)^1/2-(x+y)^1/2)). Combine all terms into a single fraction with 2y as common denominator. Multiplying the fraction by \sqrt{x-y}-\sqrt{x+y}. Divide fractions \frac{x-y}{\frac{\left(y-x\right)\left(\sqrt{x-y}-\sqrt{x+y}\right)}{2y}} with Keep, Change, Flip: a\div \frac{b}{c}=\frac{a}{1}\div\frac{b}{c}=\frac{a}{1}\times\frac{c}{b}=\frac{a\cdot c}{b}. Simplify \frac{2\left(x-y\right)y}{\left(y-x\right)\left(\sqrt{x-y}-\sqrt{x+y}\right)} multiplying the denominator by -1.

Final Answer

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

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 (x+-1y)/(0.5+-1x/2y)((x+-1y)^0.5+-1(x+y)^0.5) using the product ruleFind derivative of (x+-1y)/(0.5+-1x/2y)((x+-1y)^0.5+-1(x+y)^0.5) using the quotient ruleFind derivative of (x+-1y)/(0.5+-1x/2y)((x+-1y)^0.5+-1(x+y)^0.5) using logarithmic differentiationFind derivative of (x+-1y)/(0.5+-1x/2y)((x+-1y)^0.5+-1(x+y)^0.5) using the definition

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Function Plot

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

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1
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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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