Relation between slope of a function and its derivative

Question

What is the relation between the slope of a function and the derivative of that function? I was introduced to the idea of derivatives, which is basically the the concept of rise and run, the increase in value of the function in y axis over the increase of value on x axis. But it's the same for slopes as well right? So can I say that the derivative of a function at a point is basically the slope of the function itself?

Answer 1

The derivative is the slope of the tangent line to a function at a certain point. For example, the derivative of $f(x)=x^2$ is $f'(x)=2x$, so at $x$-value $k$ the slope of the line tangent to $f(x)$ at $x=k$ is $f'(k)=2k$.

We generally use slope for lines and derivatives for more complex curves. However, they are really just the same thing. The derivative of a line $y=ax+b$ is $y'=a$, which is the slope i.e., a line has a constant derivative. But for functions that do not have a constant slope, we define the derivative in terms of the $x$-value, such as the above example with $f(x)=x^2$.

Another way to think about derivatives and slopes is that they are the rate of change of the function at a certain point. A line describes motion at a constant rate, so its slope is constant. Other functions do not have a constant rate of change, so their slope(derivative) is defined in terms of $x$.

tl;dr, they are the same, but slopes are more commonly used with lines and derivatives with functions that do not have a constant rate of change, but they are describing the same thing.

Answer 2

The derivative can be a lot. It has a direction in which we measure a change, a location at which we do it, and what makes the derivative a local quantity, a function of which we measure a change, and finally a slope which is the change. So it depends on what we put focus on. My personal favorite of looking at it is by the Weierstraß formula: $$ f(x_0+v)=f(x_0)+ D_{x_0}(f)\cdot v + r(v) $$ where $x_0$ is the location, $f$ the (differentiable) function, $v$ the direction, $f'(x_0)=D_{x_0}(f)$ the derivative, i.e. the linear approximation at $f,$ the tangent, and $r(v)$ the error we make by such an approximation. The remainder may not have a linear term for otherwise this would add to the already given linear term $D_{x_0}(f)\cdot v.$ This condition reads thus $$ \lim_{v \to 0}\dfrac{r(v)}{\|v\|}=0 $$ I like that definition because it contains all relevant quantities, and their relevant properties, especially the linearity of $D_{x_0}(f),$ and allows generalizations in all directions, e.g. tangent bundles.

In that setup, we get the derivative $D_{x_0}(f)$ as a function of location $x_0,$ or as an operator $D_{x_0}$ on the function, and last but not least $D_{x_0}(f)\cdot v$ as the slope in direction $v.$

It is a good exercise to show that Weierstraß's formula is equivalent to the common definition via limits (a sequence of secants that converge to a tangent), and to check those quantities at an easy example like $f(x)=x^3+x.$ The crucial point is, that we need all these perspectives in physics and all are usually called just a derivative.