We are given these equations to consider:
1. x = st
2. y = e^(st)
3. z = t^2
4. w = x*y + 2*y*z + 2*x*z
Here, we will find the partial derivatives of the function w with respect to the variables s and t. Let's begin.
The first step is to compute the partial derivative of w with respect to s (i.e. ∂w/∂s). This involves applying the chain rule of differentiation to each term in the given function w.
The chain rule for differentiation states that the derivative of a composition of functions is the derivative of the outer function multiplied by the derivative of the inner function.
Using this rule, we obtain the partial derivative dw/ds, so now we have the following expression: ∂w/∂s = x*y + 2*y*z + 2*x*z
Now, we substitute the expressions for x, y, and z given above into this equation:
∂w/∂s = (s*t)*exp(s*t) + 2*exp(s*t)*t^2 + 2*s*t*t^2
The next step will involve substituting the values s=-5 and t=3 into this expression. After performing this substitution, we find that ∂w/∂s at the point (-5,3) evaluates to 12*exp(-15) + 54.
Next, we compute the partial derivative of the function w with respect to t (i.e. ∂w/∂t). This again involves application of the chain rule of differentiation to each term in the function w.
So we have this equation: ∂w/∂t = (s*t)*exp(s*t) + 2*exp(s*t)*t^2 + 2*s*t*t^2
Again, we substitute the expressions for x, y, and z into this equation:
∂w/∂t = (s*t)*exp(s*t) + 2*exp(s*t)*t^2 + 2*s*t*t^2
Finally, we substitute the values s = -5 and t =3 into this expression, which gives us the following value: ∂w/∂t at the point (-5,3) evaluates to -270 - 8*exp(-15).
In conclusion, after computing and evaluating at given points, we have:
∂w/∂s(-5,3) = 12*exp(-15) + 54
∂w/∂t(-5,3) = -270 - 8*exp(-15)