How do you us implicit differentiation to solve x( y + 2 )^5 = 8
x( y + 2 )^5 = 8
( y + 2 )^5 d/dx (x) + x d/dx(y+2)^5=d/dx 8
(y+2)^5 + 5x(y+2)^4 dy/dx=0
solve for dx/dy
(Broken Link Removed)
I assume you want dy/dx.
Differentiate both sides of the equation with respect to x. You do not have to solve explicity for y first.
(y+2)^5 + 5x (y+2)^4 *dy/dx= 0
dy/dx = -(y+2)/(5x)
Thanks.
Thanks.
To use implicit differentiation to solve the equation x(y + 2)^5 = 8 and find dy/dx, we follow these steps:
Step 1: Differentiate both sides of the equation with respect to x. Treat y as a function of x and use the chain rule to differentiate (y + 2)^5.
Differentiating x(y + 2)^5 = 8 with respect to x, we get:
d/dx [x(y + 2)^5] = d/dx [8]
Step 2: Apply the product rule to differentiate x and (y + 2)^5 separately.
Using the product rule, we have:
[(y + 2)^5] * dx/dx + x * [(d/dx)(y + 2)^5] = 0
Simplifying this expression, we have:
(y + 2)^5 + 5x(y + 2)^4 * dy/dx = 0
Step 3: Solve for dy/dx.
To solve for dy/dx (the derivative of y with respect to x), isolate dy/dx on one side of the equation. In this case, we can move the term (y + 2)^5 to the other side.
(y + 2)^5 + 5x(y + 2)^4 * dy/dx = 0
Subtracting (y + 2)^5 from both sides, we get:
5x(y + 2)^4 * dy/dx = -(y + 2)^5
Now, divide both sides by 5x(y + 2)^4 to isolate dy/dx:
dy/dx = -(y + 2)^5 / (5x * (y + 2)^4)
Simplifying the expression, we get:
dy/dx = -(y + 2) / (5x)
So, the derivative dy/dx, using implicit differentiation, is -(y + 2) / (5x).