Find dy/dx
y = (x+y)/(x√2)
I know that you can simplify the problem by arranging it to:
y = (x)/(x√2) + (y)/(x√2)
but I don't know where to go from here
I am not certain what you mean by x√2.
Please explain
apply quotient rule:
http://www.mathwords.com/d/derivative_rules.htm
y'=[(1+y')(x√2)-(x+y)(√2)]/(x√2)^2
implicit differentiation:
http://www.cliffsnotes.com/study_guide/Implicit-Differentiation.topicArticleId-39909,articleId-39882.html
solve for y'
I agree with Bob. A clarification is necessary.
(x)/(x√2) looks like 1/√2 to me, and the derivative of that term is zero.
To find dy/dx, you need to differentiate both sides of the equation with respect to x. Let's start by differentiating the equation term by term.
For the term (x)/(x√2), we can simplify it further by canceling out the x terms. This gives us 1/√2.
For the second term (y)/(x√2), we have a quotient of two functions. To differentiate this, we will use the quotient rule.
The quotient rule states that if you have a function u(x) divided by a function v(x), then the derivative of this quotient is given by:
(d/dx) [u(x)/v(x)] = [v(x) * d(u(x))/dx - u(x) * d(v(x))/dx] / [v(x)]^2
Applying this rule, let's differentiate the second term (y)/(x√2):
d/dx [(y)/(x√2)] = [x√2 * d(y)/dx - y * d(x√2)/dx] / [x√2]^2
To evaluate the derivatives, we need to find the derivatives of y with respect to x and d(x√2)/dx.
Since we are looking for dy/dx, we will isolate d(y)/dx.
Rearranging the equation, we have:
x√2 * d(y)/dx = d(x√2)/dx * y - y * d(x√2)/dx
Now, let's solve for d(y)/dx:
d(y)/dx = [d(x√2)/dx * y - y * d(x√2)/dx] / (x√2)
To find d(x√2)/dx, we can differentiate x√2 using the product rule:
d(x√2)/dx = (d(x)/dx) * √2 + x * (d(√2)/dx)
The derivative of √2 is 0 since it is a constant, and d(x)/dx is simply 1.
Therefore, d(x√2)/dx = 1 * √2 = √2
Now, we substitute this value back into the equation:
d(y)/dx = [√2 * y - y * √2] / (x√2)
Simplifying further:
d(y)/dx = 0
Therefore, the derivative of y with respect to x (dy/dx) is 0.
So, dy/dx = 0.