Use the process of implicit differentiation to find
dy/dx given that x^3e^y-ye^x=0
x^3e^y-ye^x=0
Using the product and chain rules,
3x^2 e^y + x^3 e^y y' - e^x y - e^x y' = 0
Now just solve for y'.
Ok. How to solve it that's my issue?
oh come on. The calculus is done. The rest is just algebra I.
(x^3 e^y - e^x) y' = e^x y - 3x^2 e^y
Now just divide to get y'.
Divide by 0?
huh? HUH?
y' = (e^x y - 3x^2 e^y) / (x^3 e^y - e^x)
Looks like you need to review implicit differentiation. In general, y' will be an expression involving both x and y.
To find dy/dx using the process of implicit differentiation, follow these steps:
Step 1: Write down the given equation:
x^3e^y - ye^x = 0
Step 2: Differentiate both sides of the equation with respect to x, treating y as an implicit function of x. For example, when differentiating x^3e^y with respect to x, you'll need to apply the product rule.
Let's differentiate each term separately:
Term 1: Differentiate x^3e^y with respect to x.
Using the product rule, the derivative of x^3e^y with respect to x is:
d/dx(x^3e^y) = 3x^2e^y + x^3e^y * dy/dx
Term 2: Differentiate -ye^x with respect to x.
The derivative of -ye^x with respect to x is:
d/dx(-ye^x) = -y * e^x + e^x * dy/dx
Step 3: Set the differentiated equation equal to zero to solve for dy/dx.
Combining the derivatives from step 2 and setting it equal to zero, we have:
3x^2e^y + x^3e^y * dy/dx - y * e^x + e^x * dy/dx = 0
Step 4: Collect all terms involving dy/dx on one side of the equation and all other terms on the other side.
Rearranging the equation, we have:
x^3e^y * dy/dx + e^x * dy/dx = y * e^x - 3x^2e^y
Step 5: Factor out dy/dx.
Factoring out dy/dx, we get:
(dy/dx)(x^3e^y + e^x) = y * e^x - 3x^2e^y
Step 6: Solve for dy/dx by dividing both sides by (x^3e^y + e^x).
Dividing both sides by (x^3e^y + e^x), we have:
dy/dx = (y * e^x - 3x^2e^y) / (x^3e^y + e^x)
So, the derivative dy/dx using implicit differentiation is:
dy/dx = (y * e^x - 3x^2e^y) / (x^3e^y + e^x)