# Slope of a nonlinear curve at a single point

This part of my microeconomics lesson plan has me baffled.

Consider for example the nonlinear continuous and differentiable function Y = f(X) = X 2 + 4. Suppose we want to know its slope at the point (X, Y) = (3, 13). The derivative of this function is f ’(X) = 2X, which takes on the value 6 when X = 3. Hence, the slope of this function is 6 at the point (3, 13).

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I've bold-faced the parts which I believe are disrupting my understanding, specifically. For one thing, I've always understood slope to be between two points. I can conceptualize the slope at a single point on a curve (a stick against a ball would accomplish the same - only one solution for each point) but not what it's purpose would be.

The other thing confusing me is where the derivative comes from. Each equation I've found (I did internet research on this) seems to pull a derivative from thin air, unrelated to the equation itself.

I've gotten as far as trigonometry in my schooling and each internet result I have found leads me into the realm of calculus. But that class wasn't a prerequisite for microeconomics! Can anyone shed some light on this for me? I know I'm supposed to be finding the tangent line on that point, but is there a more math-work way of accomplishing this than the visual method of plotting the graph and using a ruler?

The derivative measures the slope of a tangent at a point $P$ of the curve. That is approximately the slope of a line segment $PP'$ with $P'$ close to $P$. The closer $P'$ is chosen to $P$, the better the approximation. Thus the derivative measures the influence of changes - in principle only of very very small changes, but in practice also of reasonably sized changes (as long as the nonlinear shape of the curve does not kick in).
Finding the derivative of an elementary function (i.e. virtually anything you are able to write down in an expression) is possible using a handful of rules, such as $\frac d{dx} x^n=nx^{n-1}$, $\frac d{dx} (f+g)(x)=\frac d{dx} f(x) + \frac d{dx} g(x)$ and $\frac d{dx} (f\cdot g)(x) = f(x) \frac d{dx} g(x) + g(x) \frac d{dx} f(x)$. These should be covered in any introductory course on calculus ...