Final answer:
Implicit differentiation on the given equation e^x - y = xy^3 + e^2 - 18 and solving for dy/dx at the point (2,2) yields a value of 1, corresponding to option c).
Step-by-step explanation:
The question asks us to find the value of dy/dx at the point (2,2) for the given equation ex - y = xy3 + e2 - 18. To find dy/dx, we need to perform implicit differentiation on both sides of the equation with respect to x.
First, differentiate ex: dex/dx = ex.
Next, differentiate -y (using the chain rule as y is a function of x): -dy/dx.
On the right side, for xy3, use the product rule: d/dx(xy3) = y3 + 3xy2(dy/dx), since y is a function of x.
The derivative of a constant e2 is 0, and the derivative of -18 is also 0.
Combining all the derived parts we get the equation ex - dy/dx = y3 + 3xy2(dy/dx). At the point (2,2), we substitute x = 2 and y = 2 into this equation and solve for dy/dx.
The final equation at the point (2,2) becomes e2 - dy/dx = 23 + 3*2*22(dy/dx). Solving for dy/dx yields 1, thus, the value of dy/dx at the point (2,2) is 1, which corresponds to option c).