Chapter 2, Problem 1RCC

To determine

To express: The slope of the tangent line to the curve $y=f\left(x\right)$ at the point $\left(a,f\left(a\right)\right)$.

Explanation of Solution

Result used:

Derivative rule:

Let I be an interval, $c\in I$, and $f:I\to ℝ$ then

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

Calculation:

The curve is $y=f\left(x\right)$ then the slope of the curve is $\frac{dy}{dx}$

Given point $\left(x_0,y_0\right)=\left(a,f\left(a\right)\right)$ then $y_0=f\left(a\right)$

${y^{\prime }}_0=f^{\prime }\left(a\right)$

By the derivative rule,

${y^{\prime }}_0=\underset{x\to x_0}{\lim }\frac{f\left(x\right)-f\left(x_0\right)}{x-x_0}$

Differential function ${y^{\prime }}_0$ at the point $\left(a,f\left(a\right)\right)$ is

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

Thus slope of the tangent at $\left(a,f\left(a\right)\right)$$\underset{\_}{=f^{\prime }\left(a\right)=\underset{x\to a}{\lim }\frac{f\left(x\right)-f\left(a\right)}{x-a}}$.