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Find the implicit derivative $\frac{d}{dx}\left(-x^2+\ln\left(1+xy^2\right)+\sqrt{x^2+y}\right)=y^{5x^3}$

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Solution

Final answer to the problem

$-2x+\frac{y^2}{1+xy^2}+2\frac{1}{1+xy^2}xy\cdot y^{\prime}+\frac{1}{2\sqrt{x^2+y}}\left(2x+y^{\prime}\right)=y^{5x^3}$
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Step-by-step Solution

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The derivative of a sum of two or more functions is the sum of the derivatives of each function

$\frac{d}{dx}\left(-x^2\right)+\frac{d}{dx}\left(\ln\left(1+xy^2\right)\right)+\frac{d}{dx}\left(\sqrt{x^2+y}\right)=y^{5x^3}$

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$\frac{d}{dx}\left(-x^2\right)+\frac{d}{dx}\left(\ln\left(1+xy^2\right)\right)+\frac{d}{dx}\left(\sqrt{x^2+y}\right)=y^{5x^3}$

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Learn how to solve problems step by step online. Find the implicit derivative d/dx(-x^2+ln(1+xy^2)(x^2+y)^(1/2))=y^(5x^3). The derivative of a sum of two or more functions is the sum of the derivatives of each function. The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function. 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 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

$-2x+\frac{y^2}{1+xy^2}+2\frac{1}{1+xy^2}xy\cdot y^{\prime}+\frac{1}{2\sqrt{x^2+y}}\left(2x+y^{\prime}\right)=y^{5x^3}$

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Function Plot

Plotting: $-2x+\frac{y^2}{1+xy^2}+2\frac{1}{1+xy^2}xy\cdot y^{\prime}+\frac{1}{2\sqrt{x^2+y}}\left(2x+y^{\prime}\right)=y^{5x^3}$

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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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