Follow the steps below for the given function. (Do not use mixed numbers in your answers.)
2x + 5y = 1
Solve the equation for y.
y =
Differentiate this equation with respect to x.
y ' =
Complete the steps below to implicitly take the derivative of the original equation.
2x + 5y = 1
+ (dy)/(dx) = 0
(dy)/(dx) =
(dy)/(dx) =
so what do you not get? If you can take a derivative, this should be a snap.
To solve the given equation 2x + 5y = 1 for y, you need to isolate y. Here are the steps:
1. Start with the equation: 2x + 5y = 1
2. Subtract 2x from both sides to isolate the term with y: 5y = 1 - 2x
3. Divide both sides by 5 to solve for y: y = (1 - 2x)/5
So the solution to the equation 2x + 5y = 1 for y is y = (1 - 2x)/5.
Now, let's differentiate the equation with respect to x. To differentiate a function implicitly, you treat y as a function of x and differentiate both sides of the equation with respect to x. Here are the steps:
1. Start with the equation: 2x + 5y = 1
2. Differentiate both sides with respect to x: d(2x)/dx + d(5y)/dx = d(1)/dx
3. Since x is an independent variable, d(2x)/dx = 2 and d(1)/dx = 0.
4. For d(5y)/dx, you need to differentiate y implicitly. Since y is a function of x, you apply the chain rule by multiplying dy/dx with d(5y)/dy.
5. Applying the chain rule, d(5y)/dx = 5(dy/dx)
6. So the equation becomes: 2 + 5(dy/dx) = 0
7. Solve for dy/dx: dy/dx = -2/5
Therefore, the derivative of the equation 2x + 5y = 1 with respect to x is dy/dx = -2/5.