find dy/dx by implicit differentiation
a) xy + x^5y^2 + 3x^3 - 4 = 1
b) lny = cos x
thank you
dy/dx = (1y)+ (x)dy/dx + 5x^4y^2 + 2x^5y(dy/dx) + 9x^2
-y - 5x^4y^2 - 9x^2 = x(dy/dx) + 2x^5y (dydx)
(-y -5x^4y^2 -9x^2)/ (x + 2x^5y) = dy/dx
hi, is that all for a? please let me know, thank you Maya
To find dy/dx by implicit differentiation, you will need to differentiate both sides of the equation with respect to x and then solve for dy/dx. Here's how you can do it for each equation:
a) xy + x^5y^2 + 3x^3 - 4 = 1
Step 1: Differentiate both sides of the equation with respect to x:
d/dx(xy) + d/dx(x^5y^2) + d/dx(3x^3) - d/dx(4) = d/dx(1)
Step 2: Apply the derivative rules:
y + x(dy/dx) + 5x^4y^2 + 2x^5y(dy/dx) + 9x^2 - 0 = 0
Step 3: Rearrange the equation to solve for dy/dx:
x(dy/dx) + 2x^5y(dy/dx) = -y - 5x^4y^2 - 9x^2
Step 4: Factor out dy/dx:
(dy/dx)(x + 2x^5y) = -y - 5x^4y^2 - 9x^2
Step 5: Solve for dy/dx by dividing both sides by (x + 2x^5y):
dy/dx = (-y - 5x^4y^2 - 9x^2) / (x + 2x^5y)
Therefore, dy/dx for the equation xy + x^5y^2 + 3x^3 - 4 = 1 is (-y - 5x^4y^2 - 9x^2) / (x + 2x^5y).
b) lny = cos x
Step 1: Differentiate both sides of the equation with respect to x:
d/dx(lny) = d/dx(cos x).
Step 2: Apply the derivative rules:
(1/y)(dy/dx) = -sin x.
Step 3: Rearrange the equation to solve for dy/dx:
dy/dx = -sin x * y.
Therefore, dy/dx for the equation lny = cos x is -sin x * y.
Please note that implicit differentiation can sometimes lead to implicit solutions for dy/dx, which means they are not expressed in terms of y and x only.