exercise\npower series representation for the function. determine the radius of convergence, r. (give your…

exercise\npower series representation for the function. determine the radius of convergence, r. (give your power series representation centered at x = 0.)\nf(x)=ln(6 - x)\n\nit first recognize that f(x)=ln(6 - x) is an anti - derivative of a more familiar function. to find this function, we find d/dxln(6 - x) =

exercise\npower series representation for the function. determine the radius of convergence, r. (give your power series representation centered at x = 0.)\nf(x)=ln(6 - x)\n\nit first recognize that f(x)=ln(6 - x) is an anti - derivative of a more familiar function. to find this function, we find d/dxln(6 - x) =

Answer

Explanation:

Step1: Apply chain - rule for differentiation

Let $u = 6 - x$. Then $\frac{d}{dx}\ln(6 - x)=\frac{d}{du}\ln(u)\cdot\frac{du}{dx}$. Since $\frac{d}{du}\ln(u)=\frac{1}{u}$ and $\frac{du}{dx}=- 1$, we have $\frac{d}{dx}\ln(6 - x)=\frac{-1}{6 - x}=-\frac{1}{6(1-\frac{x}{6})}$.

Answer:

$-\frac{1}{6 - x}$