Question

Use the limit definition of the derivative to find the slope of the tangent line to the curve

Answer 1

EXPLANATION

The given function is defined by

$\begin{gathered} f(x)=6x-5 \\ at\text{ x= -4} \end{gathered}$

the slope of a tangent line, the instantaneous rate of change of a function

$f^{\prime}(x)_{\lim _{\square}x\Rightarrow a}=\frac{f(x)-f(a)}{x-a}$

Thus, we will have

$\begin{gathered} a=-4 \\ f^{\prime}(x)=\frac{6x-5-(6\times-4-5)}{x-(-4)}=\frac{6x-5+29}{x+4}=\frac{6x-24}{x+4} \\ as\text{ } \\ \lim _{x=a} \\ \text{Applying L'hopital rule} \\ f^{\prime}(x)=\frac{6}{1}=6 \end{gathered}$

Therefore, the slope will be 6