How do you find the slope of a curve at a given point?
Category: Derivatives »
Course: Calculus »
Tangent SlopesOne of the major purposes of Calculus is to find the slope of a curvy function at a specific point. Since the function is indeed curvy, and the slope is ever-changing, we often refer to the instantaneous slope at a given $x$ value as the tangent slope of the curve at that point.Ultimately, we cannot simply "eyeball" the answer. We have to use Calculus. The tangent slope of a function at a specified point is equal to the value of the function's derivative function at that $x$ value. Depending on the instructions we are given, we can get this via two approaches.
The Easy WayIf your instructions don't specify a method, you will of course choose the easier of the two ways. This method has two major steps, both of which are typically fast calculations:
- For your given $f(x)$, find $f'(x)$, which is the derivative function of $f(x)$.
- Evaluate $f'(x)$ at the $x$ value that you are interested in. In other words, if you want the derivative of $f$ at $x=a$, evaluate $f'(a)$.
The Long WayIf (and really, only if) your instructions require you to find the tangent slope using the "limit definition of the derivative" or just "the definition of the derivative" or the "limit of the difference quotient", you are being asked to do something a little more specific. It yields the same answer as the "Easy Way" above, but the method and calculation is the long form limit definition of how $f'(a)$ is rigorously defined.The major steps in this case are
- Write down $f(x)$.
- Under that, write down and simplify $f(x+h)$.
- Plug both $f(x)$ and $f(x+h)$ into the limit definition of the derivative formula, and evaluate the limit by simplifying the fraction.