Question

question
find the slope of the tangent line to the graph of (f(x)=-x - 7) at (x = - 2).
provide your answer below.
(m_{tan}=square)

Explanation:

Step1: Recall derivative formula

The derivative of a linear - function $y = ax + b$ is $y^\prime=a$. For $f(x)=-x - 7$, using the power - rule $\frac{d}{dx}(x^n)=nx^{n - 1}$, where for $y=-x-7=-1x^1-7$, the derivative $f^\prime(x)=\frac{d}{dx}(-x - 7)$.

Step2: Calculate the derivative

$\frac{d}{dx}(-x - 7)=\frac{d}{dx}(-x)+\frac{d}{dx}(-7)$. Since $\frac{d}{dx}(-x)=-1$ and $\frac{d}{dx}(-7) = 0$ (the derivative of a constant is 0), then $f^\prime(x)=-1$.

Step3: Determine the slope of the tangent line

The slope of the tangent line to the graph of $y = f(x)$ at any point $x$ is given by $f^\prime(x)$. At $x=-2$, the slope $m=f^\prime(-2)$. Since $f^\prime(x)$ is a constant function equal to $-1$ for all $x$, then $m=-1$.

Answer:

$-1$