Step-by-step Solution

Find the implicit derivative $\frac{d}{dx}\left(x\sqrt{1+y}+\sqrt{y\left(1+2x\right)}=2x\right)$

Go!
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
Dx
|◻|
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

Step-by-step explanation

Problem to solve:

$\frac{d}{dx}\left(x\sqrt{1+y}+\sqrt{y\cdot\left(1+2x\right)}=2x\right)$

Learn how to solve implicit differentiation problems step by step online.

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

Unlock this full step-by-step solution!

Learn how to solve implicit differentiation problems step by step online. Find the implicit derivative (d/dx)(x(1+y)^0.5+(y(1+2x))^0.5=2x). Apply implicit differentiation by taking the derivative of both sides of the equation with respect to the differentiation variable. The power of a product is equal to the product of it's factors raised to the same power. The derivative of a function multiplied by a constant (2) is equal to the constant times the derivative of the function. The derivative of the linear function is equal to 1.

Final Answer

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

Problem Analysis

$\frac{d}{dx}\left(x\sqrt{1+y}+\sqrt{y\cdot\left(1+2x\right)}=2x\right)$

Related formulas:

6. See formulas

Time to solve it:

~ 0.2 seconds