5–8. Calculate dy/dx using implicit differentiation.
e^y-e^x = C, where C is constant

Verified step by step guidance

1. Start by differentiating both sides of the equation with respect to x. The equation is e^y - e^x = C. Since C is a constant, its derivative is 0.
2. Differentiate e^y with respect to x. Use the chain rule: the derivative of e^y with respect to y is e^y, and then multiply by dy/dx because y is a function of x.
3. Differentiate e^x with respect to x. The derivative of e^x with respect to x is simply e^x.
4. Set up the equation from the derivatives: e^y * (dy/dx) - e^x = 0.
5. Solve for dy/dx by isolating it on one side of the equation: e^y * (dy/dx) = e^x, then divide both sides by e^y to get dy/dx = e^x / e^y.

Key Concepts

Implicit Differentiation

Implicit differentiation is a technique used to find the derivative of a function when it is not explicitly solved for one variable in terms of another. Instead of isolating y, we differentiate both sides of the equation with respect to x, applying the chain rule when differentiating terms involving y. This method is particularly useful for equations where y cannot be easily expressed as a function of x.

Chain Rule

The chain rule is a fundamental principle in calculus that allows us to differentiate composite functions. It states that if a function y is defined as a function of u, which in turn is a function of x, then the derivative of y with respect to x can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to x. This is essential in implicit differentiation when dealing with terms involving y.

Exponential Functions

Exponential functions are mathematical functions of the form f(x) = e^x, where e is the base of the natural logarithm. These functions have unique properties, such as their derivative being equal to the function itself. In the context of the given equation, understanding how to differentiate e^y and e^x is crucial for applying implicit differentiation correctly and finding dy/dx.