find dy/dx of
xsiny-ycosx=1
use the product rule....
x(siny) - y(cosx) = 1
x(cosy)dy/dx + siny - y(-sinx) - cosx(dy/dx) = 0
dy/dx(xcosy - cosx) = -siny - ysinx
dy/dx = -(siny + ysinx)/(xcosy - cosx)
To find the derivative of the given equation, we will use implicit differentiation.
Step 1: Differentiate both sides of the equation with respect to x.
d/dx(x siny - y cosx) = d/dx(1)
Step 2: Apply the product rule to differentiate the terms on the left side.
(d/dx(x))(siny) + x(d/dx(siny)) - (d/dx(y))(cosx) - y(d/dx(cosx)) = 0
Step 3: Simplify the derivative of the trigonometric functions using basic differentiation rules.
1(1) + x(cosy)(dy/dx) - (dy/dx)(cosx) - y(-sinx) = 0
Step 4: Rearrange the equation to solve for dy/dx, which is our goal.
1 + xcosy(dy/dx) - cosx(dy/dx) + y sinx = 0
Step 5: Group the terms involving dy/dx and factor out dy/dx.
(ycosx - xsiny) dy/dx = cosx - 1
Step 6: Divide both sides of the equation by (ycosx - xsiny) to isolate dy/dx.
dy/dx = (cosx - 1) / (ycosx - xsiny)
Therefore, the derivative (dy/dx) of the given equation is (cosx - 1) / (ycosx - xsiny).
To find the derivative of the given equation, we can apply implicit differentiation. Let's go step by step:
1. Start with the given equation:
xsin(y) - ycos(x) = 1
2. Differentiate both sides of the equation with respect to x using the chain rule for the trigonometric functions:
(d/dx)(xsin(y)) - (d/dx)(ycos(x)) = (d/dx)(1)
3. Differentiate each term on the left-hand side:
x(d/dx)(sin(y)) + sin(y) - (d/dx)(y)(cos(x)) - y(-sin(x)) = 0
4. Evaluate the derivatives of the trigonometric functions:
x(d/dx)(sin(y)) + sin(y) - y(-sin(x))cos(x) + ysin(x) = 0
5. Rearrange the equation to solve for the derivative (dy/dx):
x(d/dx)(sin(y)) - ycos(x)(d/dx)(sin(x)) = -sin(y) + ysin(x) - sin(x)
6. Use the known derivatives of the trigonometric functions:
x(cos(y))(dy/dx) + ycos(x)(cos(y)) = -sin(y) + ysin(x) - sin(x)
7. Solve the equation for (dy/dx):
dy/dx = (-sin(y) + ysin(x) - sin(x)) / (x(cos(y)) + ycos(x)(cos(y)))
Therefore, the derivative of the equation xsin(y) - ycos(x) = 1 with respect to x is given by:
dy/dx = (-sin(y) + ysin(x) - sin(x)) / (x(cos(y)) + ycos(x)(cos(y)))