Question

What is the relationship between the derivative of a function $f$at a point $x=c$, the slope of the tangent line to the graph of $f$at $x=c$, and the instantaneous rate of change of $f$at $x=c$?

Short answer

The derivative of $f$at $x=c$, the slope of the tangent line at $x=c$and instantaneous rate of change at $x=c$all represent the same value i.e.$\frac{f(c+h)-f\left(c\right)}{h}$.

Step-by-step solution

Step 1. Given information

A function$f\left(x\right)$.

Step 2. Explanation

The derivative of a function at any two instantaneous points $x=c$and $x=c+h$is given by $\frac{f(c+h)-f\left(c\right)}{h}$.

While the slope of the tangent line at $x=c$is also the rate of change of the function $f$for two instantaneous points $x=c$and $x=c+h$.

Therefore, the slope of the tangent line is $\frac{f(c+h)-f\left(c\right)}{h}$.

Thus the derivative of $f$at $x=c$, the slope of the tangent line at $x=c$and instantaneous rate of change at$x=c$all represent the same value.