Find dy/dx in terms of x and y if arcsin(x^5y)=xy^5.
arcsin(x^5y) = xy^5
1/√(1-x^10y^2) * (5x^4y + x^5 y') = y^5 + 5xy^4 y'
y' =
y^5 - 5x^4y/√(1-x^10y^2)
--------------------------
x^5/√(1-x^10y^2) - 5xy^4
you can massage that if you want to get rid of the fractions
To find dy/dx in terms of x and y, we can differentiate both sides of the equation with respect to x using implicit differentiation.
Let's start by differentiating the equation arcsin(x^5y) = xy^5 with respect to x.
Differentiating the left side:
d/dx(arcsin(x^5y))
Using the chain rule:
d/dx(arcsin(u)) = 1/sqrt(1 - u^2) * du/dx
Let u = x^5y, so du/dx = 5x^4y + x^5 * dy/dx
Differentiating the right side:
d/dx(xy^5) = y^5 + x * d/dx(y^5)
Using the power rule:
d/dx(y^n) = n * y^(n-1) * dy/dx
So, d/dx(xy^5) = y^5 + 5xy^4 * dy/dx
Now, equating the two sides and substituting the derivatives obtained:
1/sqrt(1 - u^2) * (5x^4y + x^5 * dy/dx) = y^5 + 5xy^4 * dy/dx
Multiplying both sides by sqrt(1 - u^2):
5x^4y + x^5 * dy/dx = (y^5 + 5xy^4 * dy/dx) * sqrt(1 - x^10y^2)
Rearranging the equation to solve for dy/dx:
x^5 * dy/dx - (5xy^4 * dy/dx) * sqrt(1 - x^10y^2) = y^5 - 5x^4y * sqrt(1 - x^10y^2)
Factor out dy/dx terms:
dy/dx (x^5 - 5xy^4 * sqrt(1 - x^10y^2)) = y^5 - 5x^4y * sqrt(1 - x^10y^2)
Finally, dividing both sides by (x^5 - 5xy^4 * sqrt(1 - x^10y^2)):
dy/dx = (y^5 - 5x^4y * sqrt(1 - x^10y^2)) / (x^5 - 5xy^4 * sqrt(1 - x^10y^2))
Therefore, dy/dx in terms of x and y is:
dy/dx = (y^5 - 5x^4y * sqrt(1 - x^10y^2)) / (x^5 - 5xy^4 * sqrt(1 - x^10y^2))
To find dy/dx, we will use implicit differentiation.
Step 1: Differentiate both sides of the equation with respect to x.
On the left side, we will use the chain rule. Since arcsin(x^5y) is a composition of two functions, we differentiate the outer function and multiply it by the derivative of the inner function.
d/dx(arcsin(x^5y)) = d/dx(xy^5)
The derivative of arcsin(u) with respect to u is 1/sqrt(1 - u^2), so applying the chain rule, we get:
[1/sqrt(1 - (x^5y)^2)] * d/dx(x^5y) = (y^5) * 1 + x * d/dx(y^5)
Step 2: Simplify the equation.
To simplify the equation, we need to find the derivatives of x^5y and y^5 with respect to x.
Using the product rule for x^5y, we have:
d/dx(x^5y) = d/dx(x^5) * y + x^5 * d/dx(y)
= 5x^4 * y + x^5 * dy/dx
Similarly, using the power rule for y^5, we have:
d/dx(y^5) = 5y^4 * dy/dx
Step 3: Substitute the simplified expressions into the equation.
Substituting the simplified expressions into the equation, we have:
[1/sqrt(1 - (x^5y)^2)] * (5x^4 * y + x^5 * dy/dx) = y^5 + 5xy^4 * dy/dx
Step 4: Solve for dy/dx.
To solve for dy/dx, we isolate the term by moving all the other terms to the other side of the equation:
[1/sqrt(1 - (x^5y)^2)] * (5x^4 * y + x^5 * dy/dx) - 5xy^4 * dy/dx = y^5
Now, we can factor out dy/dx as a common term:
dy/dx * [1/sqrt(1 - (x^5y)^2) * x^5 - 5xy^4] = y^5 - [1/sqrt(1 - (x^5y)^2)] * (5x^4 * y)
Finally, divide both sides of the equation by [1/sqrt(1 - (x^5y)^2) * x^5 - 5xy^4] to solve for dy/dx:
dy/dx = (y^5 - [1/sqrt(1 - (x^5y)^2)] * (5x^4 * y)) / [1/sqrt(1 - (x^5y)^2) * x^5 - 5xy^4]
So, we have found dy/dx in terms of x and y.