Question

for the linear function y = f(x) = -5x - 6.
a. find $\frac{df}{dx}$ at x = -1.
f(-1) = -5

b. find a formula for x = f^(-1)(y).
f^(-1)(y) =

c. find $\frac{df^{-1}}{dy}$ at y = f(-1).
(f^(-1))(f(-1)) =

Explanation:

Step1: Differentiate the function

The derivative of $y = f(x)=-5x - 6$ with respect to $x$ is $f'(x)=\frac{df}{dx}=-5$. The derivative of a linear function $y = ax + b$ (where $a$ and $b$ are constants) is $a$. Since $a=-5$, $f'(-1)=-5$.

Step2: Find the inverse function

Start with $y=-5x - 6$. Solve for $x$ in terms of $y$.
Add 6 to both sides: $y + 6=-5x$.
Then divide by - 5: $x=f^{-1}(y)=-\frac{y + 6}{5}=-\frac{1}{5}y-\frac{6}{5}$.

Step3: Differentiate the inverse function

The derivative of $f^{-1}(y)=-\frac{1}{5}y-\frac{6}{5}$ with respect to $y$ is $(f^{-1})'(y)=\frac{df^{-1}}{dy}=-\frac{1}{5}$. The value of $(f^{-1})'(y)$ is constant for all $y$. So, $(f^{-1})'(f(-1))=-\frac{1}{5}$.

Answer:

b. $f^{-1}(y)=-\frac{y + 6}{5}$
c. $(f^{-1})'(f(-1))=-\frac{1}{5}$