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Find the implicit derivative $\frac{d}{dx}\left(y^2=x+\ln\left(\frac{y}{x}\right)\right)$

Step-by-step Solution

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Solution

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Final Answer

$y^{\prime}=\frac{yx-y}{\left(2y^2-1\right)x}$
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Step-by-step Solution

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Apply implicit differentiation by taking the derivative of both sides of the equation with respect to the differentiation variable

$\frac{d}{dx}\left(y^2\right)=\frac{d}{dx}\left(x+\ln\left(\frac{y}{x}\right)\right)$

Learn how to solve discriminant of quadratic equation problems step by step online.

$\frac{d}{dx}\left(y^2\right)=\frac{d}{dx}\left(x+\ln\left(\frac{y}{x}\right)\right)$

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Learn how to solve discriminant of quadratic equation problems step by step online. Find the implicit derivative d/dx(y^2=x+ln(y/x)). Apply implicit differentiation by taking the derivative of both sides of the equation with respect to the differentiation variable. 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 linear function is equal to 1. The derivative of a sum of two or more functions is the sum of the derivatives of each function.

Final Answer

$y^{\prime}=\frac{yx-y}{\left(2y^2-1\right)x}$

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Function Plot

Plotting: $y^{\prime}=\frac{yx-y}{\left(2y^2-1\right)x}$

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1
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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

How to improve your answer:

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

Main Topic: Discriminant of Quadratic Equation

Quadratic equations are those algebraic equations of the form ax^2+bx+c, where a, b, and c are constant values. The discriminant of a quadratic equation is calculated using the formula D=b^2-4ac, and it helps us to determine how many roots an equation of this type has. When D>0 the equation has two real roots, when D<0 the equation has no real roots, and when D=0 the equation has a repeated real root.

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