Implicit diff.
y^2 + 4xy +4y^2=40
2yy'+ _______ + 8y*y=0
I am confused with the 4xy.
Would you write this out to be
4x*y'+4y OR
4x*4y' + 4y*4x'?
Please help
First off, I suspect a typo. Why have two y^2 terms? I figure you meant
x^2 + 4xy +4y^2=40
Just do things term by term, and don't forget the product rule:
2x + 4y + 4xy' + 8yy' = 0
y'(4x+8y) = -(2x+4y)
y' = -1/2
Weird, huh? Not if you note that
x^2 + 4xy +4y^2 is just (x+2y)^2
To find the derivative using implicit differentiation, we differentiate both sides of the equation with respect to the independent variable. Let's differentiate the given equation step by step:
Step 1: Differentiate y^2 + 4xy + 4y^2 = 40 with respect to x.
For the term y^2, we use the power rule, which states that the derivative of y^n with respect to x is n * y^(n-1) * y', where y' represents dy/dx (the derivative of y with respect to x).
Differentiating y^2, we get 2y * y'.
Step 2: The term 4xy can be viewed as 4x * y. To differentiate this term, we can use the product rule, which states that the derivative of a product of two functions u(x) and v(x), with respect to x, is u'(x) * v(x) + u(x) * v'(x).
Applying the product rule to 4xy, we get (d/dx)(4x * y) = (4 * y) * (d/dx)(x) + (d/dx)(4x) * y
= 4y + 4xy'
Step 3: Differentiating 4y^2 using the power rule, we get 8y * y'.
So, after differentiating both sides, we have:
2y * y' + 4y + 4xy' + 8y * y' = 0
Next, we can simplify the equation by combining like terms:
2yy' + 4y + 4xy' + 8yy' = 0
Now, to find the missing term in the expression 2yy' + _______ + 8yy' = 0, we add like terms with the same derivative, which in this case are 4y and 4xy'.
Hence, we can write the final equation as:
2yy' + 4y + 4xy' + 8yy' = 0
I hope this explanation clarifies how to handle the term 4xy when differentiating using implicit differentiation. Let me know if you have any further questions!