Topics

More AI tools from SnapXam

Exercise

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

Step-by-step Solution

1

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

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

The derivative of the constant function ($1$) is equal to zero

$\frac{d}{dx}\left(x^2+xy-y^2\right)=0$
3

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

Make diagrams like this with our diagram generator
$\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(xy\right)+\frac{d}{dx}\left(-y^2\right)=0$
4

The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function

$\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(xy\right)-\frac{d}{dx}\left(y^2\right)=0$
5

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=x$ and $g=y$

Make diagrams like this with our diagram generator
$\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(x\right)y+x\frac{d}{dx}\left(y\right)-\frac{d}{dx}\left(y^2\right)=0$
6

The derivative of the linear function is equal to $1$

$\frac{d}{dx}\left(x^2\right)+y+xy^{\prime}-\frac{d}{dx}\left(y^2\right)=0$
7

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$\frac{d}{dx}\left(x^2\right)+y+xy^{\prime}-2y^{1}\frac{d}{dx}\left(y\right)=0$
8

Any expression to the power of $1$ is equal to that same expression

$\frac{d}{dx}\left(x^2\right)+y+xy^{\prime}-2y\frac{d}{dx}\left(y\right)=0$
9

The derivative of the linear function is equal to $1$

$\frac{d}{dx}\left(x^2\right)+y+xy^{\prime}-2y\cdot y^{\prime}=0$
10

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

Make diagrams like this with our diagram generator
$2x+y+xy^{\prime}-2y\cdot y^{\prime}=0$
11

Group the terms of the equation by moving the terms that have the variable $y^{\prime}$ to the left side, and those that do not have it to the right side

$xy^{\prime}-2y\cdot y^{\prime}=-2x-y$
12

Factor the polynomial $xy^{\prime}-2y\cdot y^{\prime}$ by it's greatest common factor (GCF): $y^{\prime}$

$y^{\prime}\left(x-2y\right)=-2x-y$
13

Divide both sides of the equation by $x-2y$

$y^{\prime}=\frac{-2x-y}{x-2y}$

Final answer to the exercise

$y^{\prime}=\frac{-2x-y}{x-2y}$

Try other ways to solve this exercise

  • Choose an option
  • Find the derivative using the definition
  • Find the derivative using the product rule
  • Find the derivative using the quotient rule
  • Find the derivative using logarithmic differentiation
  • Find the derivative
  • Integrate by partial fractions
  • Product of Binomials with Common Term
  • FOIL Method
  • Integrate by substitution
  • Load more...
Can't find a method? Tell us so we can add it.

Similar Questions Solved

$\frac{d}{dx}\left(x^2+xy-y^2=4\right)$

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

$\frac{d}{dx}\left(2x^2+3y^2-4xy=36\right)$

$\frac{d}{dx}\left(x^3-xy+y^2=7\right)$

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

$\frac{d}{dx}\left(x^3-xy+y^2=4\right)$

$\frac{d}{dx}\left(2xy-y^2=1\right)$

Main Topic: Sum Rule of Differentiation

The sum rule is a method to find the derivative of a function that is the sum of two or more functions.

Related Topics

Symbolic 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
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

Empowering students and teachers through AI

By signing up, you agree to SnapXam's Terms and Privacy Policy.

Already have an account? Login here
Forgot your password?

Empowering students and teachers through AI



Register here if you don't have an account.
Forgot your password?

Your Personal Math Tutor. Powered by AI

Available 24/7, 365 days a year.

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

Access in depth explanations with descriptive diagrams.

Choose between multiple solving methods.

Download unlimited solutions in PDF format.

Premium access on our iOS and Android apps.

Join 1M+ students worldwide in problem solving.

Choose the plan that suits you best:
Pay $39.97 USD securely with your payment method.
Please hold while your payment is being processed.
Create an Account