considering that x is the independent variable in this equation:
y + y^3 + 3 = e^y^2 + 3^x * cos(3y) - x
Evaluate dy/dx
I get to:
dy/dx (1+3y) = e^y^2 * 3y^2 * dy/dx + 3^x * ln3 * (-sin(3y)) * 3 *dy/dx -1
Will it give me?
dy/dx(1+3y) =
dy/dx((e^y^2 * 3y^2) +
(3^x * ln3 * (-sin(3y)) * 3 ) -1
dy/dx(1+3y) =
dy/dx((e^y^2 * 3y^2) - (9^xsin(3y)ln3))
-1
1 / ((e^y^2 * 3y^2) - (9^xsin(3y)ln3) - (1+3y))
last one is supposed to be
dy/dx = 1 / ((e^y^2 * 3y^2) - (9^xsin(3y)ln3) - (1+3y))
To evaluate dy/dx for the given equation, you need to differentiate each term with respect to x.
Starting from the given equation:
y + y^3 + 3 = e^(y^2) + 3^x * cos(3y) - x
Differentiating both sides with respect to x, you will apply the chain rule for differentiation:
d/dx(y) + d/dx(y^3) + d/dx(3) = d/dx(e^(y^2)) + d/dx(3^x * cos(3y)) - d/dx(x)
The derivative of a constant term, such as 3, with respect to x is 0.
The derivative of y with respect to x is dy/dx.
The derivative of y^3 with respect to x can be calculated using the chain rule:
d/dx(y^3) = 3y^2 * dy/dx
For the term e^(y^2), you need to apply the chain rule:
d/dx(e^(y^2)) = e^(y^2) * d/dx(y^2) * dy/dx
Using the product rule, the derivative of 3^x * cos(3y) with respect to x is:
d/dx(3^x * cos(3y)) = ln(3) * 3^x * cos(3y) * dy/dx
The derivative of x with respect to x is simply 1.
Putting all the derivatives together, you get:
dy/dx + 3y^2 * dy/dx + 0 = e^(y^2) * 2y * dy/dx + ln(3) * 3^x * cos(3y) * dy/dx - 1
Combining like terms, you end up with:
dy/dx(1 + 3y) = e^(y^2) * 2y * dy/dx + ln(3) * 3^x * cos(3y) * dy/dx - 1
Now, just solve for dy/dx by dividing both sides by (1 + 3y):
dy/dx = (e^(y^2) * 2y + ln(3) * 3^x * cos(3y) - 1) / (1 + 3y)
So, the final expression for dy/dx is:
dy/dx = (e^(y^2) * 2y + ln(3) * 3^x * cos(3y) - 1) / (1 + 3y)