Chapter 14.4, Problem 7AYU

To determine

Check whether the statement is true or false:

The slope of the tangent line to a function $f$ at $\left(c,f\left(c\right)\right)$ is the derivative of $f$ at $c$.

Answer to Problem 7AYU

Solution:

True.

Explanation of Solution

Given:

The function is $y=f\left(x\right)$.

At the point $x=c$, $y=f\left(c\right)$.

The slope of the tangent line of a function $y=f\left(x\right)$ at the point $x=c$ is given by;

$m=\underset{x\to c}{\lim }\frac{f\left(x\right)-f\left(c\right)}{x-c}$

Moreover, the derivative of a function $y=f\left(x\right)$ at the point $c$ is given by;

$f^{\prime }\left(c\right)=\underset{x\to c}{\lim }\frac{f\left(x\right)-f\left(c\right)}{x-c}$

Therefore the slope of the tangent line to $f$ at $\left(c,f\left(c\right)\right)$ is the derivative of $f$ at $c$.

Hence the given statement is True.