👉 Try now NerdPal! Our new math app on iOS and Android

Find the implicit derivative $\frac{d}{dx}\left(x^2y+3xy^3=\frac{-111}{x^2}\right)$

Step-by-step Solution

Go!
Math mode
Text mode
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
|◻|
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

Final Answer

$y^{\prime}=\frac{222-2x^{4}y-3y^3x^{3}}{x^{4}\left(x+9y^2\right)}$
Got another answer? Verify it here!

Step-by-step Solution

Specify the solving method

1

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

$\frac{d}{dx}\left(x^2y+3xy^3\right)=\frac{d}{dx}\left(\frac{-111}{x^2}\right)$

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

$\frac{d}{dx}\left(x^2y+3xy^3\right)=\frac{d}{dx}\left(\frac{-111}{x^2}\right)$

Unlock unlimited step-by-step solutions and much more!

Unlock the first 2 steps of this solution.

Learn how to solve implicit differentiation problems step by step online. Find the implicit derivative d/dx(x^2y+3xy^3=-111/(x^2)). Apply implicit differentiation by taking the derivative of both sides of the equation with respect to the differentiation variable. 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}}. Simplify \left(x^2\right)^2 using the power of a power property: \left(a^m\right)^n=a^{m\cdot n}. In the expression, m equals 2 and n equals 2. The derivative of the constant function (-111) is equal to zero.

Final Answer

$y^{\prime}=\frac{222-2x^{4}y-3y^3x^{3}}{x^{4}\left(x+9y^2\right)}$

Give us your feedback!

Function Plot

Plotting: $y^{\prime}=\frac{222-2x^{4}y-3y^3x^{3}}{x^{4}\left(x+9y^2\right)}$

SnapXam A2
Answer Assistant

beta
Got a different answer? Verify it!

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
|◻|
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:

Main Topic: Implicit Differentiation

Implicit differentiation makes use of the chain rule to differentiate implicitly defined functions. For differentiating an implicit function y(x), defined by an equation R(x, y) = 0, it is not generally possible to solve it explicitly for y(x) and then differentiate. Instead, one can differentiate R(x, y) with respect to x and y and then solve a linear equation in dy/dx for getting explicitly the derivative in terms of x and y.

Used Formulas

7. See formulas

Your Math & Physics Tutor. Powered by AI

Available 24/7, 365.

Complete step-by-step math solutions. No ads.

Includes multiple solving methods.

Support for more than 100 math topics.

Premium access on our iOS and Android apps as well.

20% discount on online tutoring.

Choose your Subscription plan:
Have a promo code?
Pay $39.97 USD securely with your payment method.
Please hold while your payment is being processed.
Create an Account