Given the equation $e^x-y=x{y}^3+e^2-18$, what is the value of $\frac{dy}{dx}$ at the point $(2,2)$?

• A. $\frac{1}{7}$
• B. $-7$
• C. $-\frac{1}{7}$
• D. $7$

Answer: (C)

Verified step by step guidance

1. Step 1: Recognize that the problem involves implicit differentiation since the equation contains both x and y variables. The goal is to find dy/dx at the given point (2, 2).
2. Step 2: Differentiate both sides of the equation e^x - y = x y^3 + e^2 - 18 with respect to x. Remember to apply the chain rule when differentiating terms involving y, as y is a function of x.
3. Step 3: For the left-hand side, differentiate e^x to get e^x and differentiate -y to get -dy/dx. For the right-hand side, use the product rule on x y^3, which gives (d/dx)(x y^3) = y^3 + x * 3y^2 * dy/dx. The derivative of constants like e^2 and -18 is 0.
4. Step 4: Combine all differentiated terms into a single equation: e^x - dy/dx = y^3 + x * 3y^2 * dy/dx. Rearrange the equation to isolate dy/dx.
5. Step 5: Substitute the point (2, 2) into the equation to evaluate dy/dx. Plug in x = 2 and y = 2, and solve for dy/dx algebraically.