Final answer:
If xy+7e^(y)=7e, the second derivative at the point where x=0 is d²y/dy² + e^y = 0.
Step-by-step explanation:
To find the second derivative at the point where x=0, we first need to differentiate the given equation with respect to y. This will give us a differential equation in terms of x and y. Let's differentiate the equation:
d/dy(xy+7e^y) = d/dy(7e)
Using the product rule and the chain rule, we get:
x(dy/dy) + y + 7e^y(dy/dy) = 0
Simplifying, we get:
x + y + 7e^y = 0
Now, let's differentiate this equation with respect to y again to find the second derivative:
d²/dy²(x + y + 7e^y) = 0
Since we are looking for the second derivative at the point where x=0, we can ignore the x term.
Differentiating the remaining terms, we get:
d²y/dy² + e^y = 0
This is the second derivative at the point where x=0.