**Differential Equations**

We know how to find derivatives of functions in the standard form. However not all functions look like that. The hyperbola is one such example of not following this rule. If we wanted to calculate the slope at say this point how would we do that? Well it is possible, but we have to use slightly more sophisticated derivatives. Let’s take a look at a simpler function such as the function $ x + y = 1$. Next let’s see how the differentiation of the function of the circle works. Take $ x^2 + y^2 = 1$ and then take the derivative. This type of equation is what we call a differential equation. Differential equations are common in calculus where you can get a formula for the slope of an equation, but need to figure out the initial equation itself.

To do this we must do the process backwards with integration. If we were given the differential equation $ \frac{dy}{dx}=xy$ and asked to find the equation of this graph how would we do it? Then we must remember to add a $ + C$ to one side of the equation. This is because we don’t know exactly where the graph is on the slope field of the equation and different starting points can give vastly different results.

- Differential Equation – an equation involving a derivative
- Antiderivative of f(x) – The function who’s derivative is equal to f(x)
- Slope Field – a graph displaying the slope of a function at specific points

**What is the slope of a parabola?**

Using differential equations what is the slope of a hyperbola centered at the origin, where $ a$ and $ b$ are $ 1$. Once you have done this graph the slope field and try to figure out what happens to the hyperbola as your $ x$ and $ y$ values get very large. Does the hyperbola approach a specific value, what happens if you change the values of $ a$ and $ b$?

*Remember to take the quiz before moving on!*

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