Find dy/dx by implicit differentation.
ysin(x^2)=-xsin(y^2)
an equilateral triangle is inscribed in a circle. suppose the radius of the circle increases at 3ft/sec. how fast is the area of the triangle increasing when the radius of the circle is 4 feet?
A little piggyback here, eh?
ysinx² = -xsiny²
y' sinx² + 2xycosx² = -1siny² -2x²cosy² y'
y'(sinx²+2x²cosy²) = -(siny² + 2xycosx²)
y' = -(siny² + 2xycosx²)/(sinx²+2x²cosy²)
To find dy/dx by implicit differentiation for the equation ysin(x^2) = -xsin(y^2), follow these steps:
Step 1: Differentiate both sides of the equation with respect to x.
d/dx (ysin(x^2)) = d/dx (-xsin(y^2))
Step 2: Apply product rule and chain rule to differentiate each term.
y * d/dx(sin(x^2)) + sin(x^2) * dy/dx = -sin(y^2) - x * d/dx(sin(y^2))
Step 3: Simplify and isolate the terms involving dy/dx.
y * d/dx(sin(x^2)) + sin(x^2) * dy/dx = -sin(y^2) - x * d/dy(sin(y^2)) * dy/dx
Step 4: Substitute the derivative of sin(x^2) and sin(y^2) into the equation.
y * 2x * cos(x^2) + sin(x^2) * dy/dx = -sin(y^2) - x * 2y * cos(y^2) * dy/dx
Step 5: Rearrange the equation to isolate dy/dx.
dy/dx * (sin(x^2) + x * 2y * cos(y^2)) = -sin(y^2) - y * 2x * cos(x^2)
Step 6: Finally, divide both sides of the equation by (sin(x^2) + x * 2y * cos(y^2)) to solve for dy/dx.
dy/dx = (-sin(y^2) - y * 2x * cos(x^2)) / (sin(x^2) + x * 2y * cos(y^2))
So, the derivative dy/dx by implicit differentiation for the given equation is (-sin(y^2) - y * 2x * cos(x^2)) / (sin(x^2) + x * 2y * cos(y^2)).