Question

Find the equation of the tangent in:f(x) = 1 / [(2x-1)⁶]x = 2

Answer 1

So,

Given the function:

`f(x)=(1)/((2x-1)^6)`

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:

`f^(\prime)(x)=(0\cdot(2x-1)^6-1\cdot6(2x-1)^5\cdot2)/(\lbrack(2x-1)^6\rbrack^2)`

If we rewrite:

`\begin{gathered} f^(\prime)(x)=(-12(2x-1)^5)/((2x-1)^(12)) \\ \\ f^(\prime)(x)=(-12)/((2x-1)^7) \end{gathered}`

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,

`f^(\prime)(2)=(-12)/((2(2)-1)^7)\to(-12)/(3^7)=-(12)/(2187)`

Then, we could use the fact that the equation of a line can be found using the following:

`y=y_1+m(x-x_1)`

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:

`f(2)=(1)/((2(2)-1)^6)=(1)/(729)`

Therefore, the point is (2 , 1/729).

Finally, we replace all these values in the equation given:

`\begin{gathered} y=(1)/(729)-(12)/(2187)(x-2) \\ \\ y=(1)/(729)-(12)/(2187)x+(24)/(2187) \\ \\ y=-(12)/(2187)x+(1)/(81) \\ \\ y=(-4)/(729)x+(1)/(81) \end{gathered}`

And that's the equation of the tangent line to that function at the point x=2.