find dy/dx (implicit differentiation) 1/x^2 + 1/y^2 = 5
(please do a step by step solution idk how to get the ans :"))
Sure, here's a step by step solution for finding dy/dx using implicit differentiation:
1. Differentiate both sides of the equation with respect to x:
d/dx (1/x^2 + 1/y^2) = d/dx (5)
2. Use the chain rule and power rule:
(-2/x^3)dx/dy + (-2/y^3)(dy/dx) = 0
3. Rearrange the terms to isolate dy/dx:
(-2/y^3)(dy/dx) = (2/x^3)(dx/dy)
4. Solve for dy/dx:
dy/dx = ((2/x^3)(dx/dy)) / (-2/y^3)
5. Substitute dx/dy = -x^2/y^2 (from the given equation):
dy/dx = ((2/x^3)(-x^2/y^2)) / (-2/y^3)
6. Simplify and cancel out terms:
dy/dx = (2y^3) / (-2x^2)
dy/dx = -y^3 / x^2
Therefore, the derivative dy/dx is -y^3/x^2.
the title suggested implicit differentiation , so ..
1/x^2 + 1/y^2 = 5
y^2 + x^2 = 5x^2 y^2
2y dy/dx + 2x = 5x^2(2y dy/dx) + y^2(10x)
2y dy/dx + 2x = 10x^2 y dy/dx + 10xy^2
2y dy/dx - 10x^2 y dy/dx = 10xy^2 - 2x
2dy/dx(y - 5x^2 y) = 10xy^2 - 2x
dy/dx = 2x(5y^2 - 1)/(2(y - 5x^2 y)
= x(5y^2 - 1) / (y(y - 5x^2) )
done the way the bot did it, which it actually got right, but was not asked for.
1/x^2 + 1/y^2 = 5
x^-2 + y^-2 = 5
-2x^-3 - 2y^-3 dy/dx = 0
2/y^3 dy/dx = -2/x^3
dy/dx = (-1/x^3) / (1/y^3)
= -y^3 / x^3
or
-(y/x)^3
just an alternate solution, looks much simpler
To find dy/dx using implicit differentiation, we need to differentiate both sides of the equation with respect to x. This means we treat y as a function of x and apply the chain rule whenever necessary.
Let's start by rewriting the equation:
1/x^2 + 1/y^2 = 5
Step 1: Differentiate both sides of the equation with respect to x.
d/dx (1/x^2) + d/dx (1/y^2) = d/dx (5)
Step 2: Apply the chain rule to differentiate 1/x^2 and 1/y^2.
-2/x^3 + 2/y^3 * (dy/dx) = 0
Step 3: Simplify the equation by combining like terms and isolating dy/dx.
-2/x^3 + 2/y^3 * (dy/dx) = 0
2/y^3 * (dy/dx) = 2/x^3
(dy/dx) = (2/x^3) / (2/y^3)
(dy/dx) = y^3/x^3
Therefore, the derivative dy/dx is equal to y^3/x^3.