Respuesta :
So,
Given the function:
[tex]f(x)=\frac{1}{(2x-1)^6}[/tex]We want to find the equation of the tangent line to this function at the point x=2.
The first thing we need to do, is to find the deritative of the function.
We could use the quotient rule as follows:
[tex]f^{\prime}(x)=\frac{0\cdot(2x-1)^6-1\cdot6(2x-1)^5\cdot2}{\lbrack(2x-1)^6\rbrack^2}[/tex]If we rewrite:
[tex]\begin{gathered} f^{\prime}(x)=\frac{-12(2x-1)^5}{(2x-1)^{12}} \\ \\ f^{\prime}(x)=\frac{-12}{(2x-1)^7} \end{gathered}[/tex]Now, we need to find the value of the deritative of this function at the point x=2. That's because that's the slope of the tangent line at that point.
So,
[tex]f^{\prime}(2)=\frac{-12}{(2(2)-1)^7}\to\frac{-12}{3^7}=-\frac{12}{2187}[/tex]Then, we could use the fact that the equation of a line can be found using the following:
[tex]y=y_1+m(x-x_1)[/tex]Where (x1,y1) is a point that lie on the line and m is the slope.
We got that x1 = 2, and the value of y1 will be the value of f(x) when x=2:
[tex]f(2)=\frac{1}{(2(2)-1)^6}=\frac{1}{729}[/tex]Therefore, the point is (2 , 1/729).
Finally, we replace all these values in the equation given:
[tex]\begin{gathered} y=\frac{1}{729}-\frac{12}{2187}(x-2) \\ \\ y=\frac{1}{729}-\frac{12}{2187}x+\frac{24}{2187} \\ \\ y=-\frac{12}{2187}x+\frac{1}{81} \\ \\ y=\frac{-4}{729}x+\frac{1}{81} \end{gathered}[/tex]And that's the equation of the tangent line to that function at the point x=2.
