What is the implicit derivative of #y=(x-y)^2+4xy+5y^2 #?

Answer 1

The implicit derivative is #dy/dx= (2x+2y)/(-12y-2x+1)#.

First, you would derive this by turning #y# into #dy/dx# when needed since it is a function while treating #x# as the variable. For the #(x-y)^2#, the chain rule is applied. This will result in:
#dy/dx=2(x-y)*(1-dy/dx)+4y+4xdy/dx+10ydy/dx#

For simplicity, we can distribute the values in the parentheses and then implement quadratic multiplication.

#dy/dx=(2x-2y)*(1-dy/dx)+4y+4xdy/dx+10ydy/dx#
#dy/dx=2x-2xdy/dx-2y+2ydy/dx+4y+4xdy/dx+10ydy/dx#

We now simplify like terms to get:

#dy/dx=2x+12ydy/dx+2y+2xdy/dx#
Move over all of the terms with #dy/dx# to the other side:
#-12ydy/dx-2xdy/dx+dy/dx=2x+2y#
Factor the #dy/dx# from the left side to get:
#dy/dx(-12y-2x+1)=2x+2y#
Divide the left and right sides by #(-12y-2x+1)# to achieve:
#dy/dx= (2x+2y)/(-12y-2x+1)#

There is your answer! Hope it helps!

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To find the implicit derivative of ( y = (x - y)^2 + 4xy + 5y^2 ), differentiate both sides of the equation with respect to ( x ).

[ \frac{d}{dx} [y] = \frac{d}{dx} [(x - y)^2 + 4xy + 5y^2] ]

Apply the chain rule and product rule where necessary.

[ \frac{dy}{dx} = \frac{d}{dx} [(x - y)^2] + \frac{d}{dx} [4xy] + \frac{d}{dx} [5y^2] ]

[ = 2(x - y) \frac{d}{dx} [x - y] + 4y + 4x\frac{dy}{dx} + 10y\frac{dy}{dx} ]

Now, differentiate ( x - y ) with respect to ( x ) to get ( \frac{d}{dx} [x - y] = 1 - \frac{dy}{dx} ).

[ \frac{dy}{dx} = 2(x - y) \left(1 - \frac{dy}{dx}\right) + 4y + 4x\frac{dy}{dx} + 10y\frac{dy}{dx} ]

Now, solve for ( \frac{dy}{dx} ) by isolating it on one side of the equation. This equation is implicit and can't be directly solved for ( \frac{dy}{dx} ).

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7