SOLUTION: Find dy/dx given that y=1+x^(2)e^(y)

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Question 223602: Find dy/dx given that y=1+x^(2)e^(y)
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
To find dy/dx we will need to understand:
  • Derivatives of a sum of functions
  • Derivatives of a product of functions
  • The Chain Rule
  • Implicit differentiation

I hope you understand most of these because I am not going to go into great detail.

We'll start by
  • Presuming that y is a function of x
  • Breaking the rest of the equation into "bite-size" functions of x. (By "bite-size" I mean functions for which we can easily find the derivative.)

So let's define u(x) = 1 and v(x) = x%5E2e%5Ey. This allows us to rewrite the equation as:
y = u + v
And the derivative is easy for sums of functions:
Equation y' = u' + v'
u' is easy but v' is not. So we will break v into smaller pieces. Let's define p%28x%29+=+x%5E2 and q%28x%29+=+e%5Ey. This makes v = p*q. And the derivative of this product of functions:
v' = p*q' + q*p'
By the Chain rule,
q' = e%5Ey%2Ay%27
and p' is an easy derivative:
p' = 2x
so by substituting these into v' = p*q' + q*p' we get:
v' = x%5E2%2Ae%5Ey*y' + e%5Ey%2A2x
Now by substituting u' (which is 0) and v' (above) into y' = u' + v' we get:
y' = 0+%2B+x%5E2%2Ae%5Ey*y' + e%5Ey%2A2x
which simplifies to:
y' = x%5E2%2Ae%5Ey*y' + 2xe%5Ey
Now all we need to do is solve for y'. Start by subtracting x%5E2%2Ae%5Ey*y' from both sides. This will gather the y' terms on the left side:
y' - x%5E2%2Ae%5Ey*y' = 2xe%5Ey
Next we'll factor out y' on the left side:
y'*%281+-+x%5E2%2Ae%5Ey%29 = 2xe%5Ey
And finally divide both sides by %281+-+x%5E2%2Ae%5Ey%29:
y' = dy/dx = %282xe%5Ey%29%2F%281+-+x%5E2%2Ae%5Ey%29