Q. 7 - Calculus

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 $x=c$?

Step-by-Step Solution

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 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.