Question

Given f(x) = (lnx)^3 find the line tangent to f at x = 3

Answer 1

Explanation

We must the tangent line at x = 3 of the function:

$f(x)=(\ln x)^3.$

The tangent line is given by:

$y=m*(x-h)+k.$

Where:

• m is the slope of the tangent line of f(x) at x = h,

,

• k = f(h) is the value of the function at x = h.

In this case, we have h = 3.

1) First, we compute the derivative of f(x):

$f^{\prime}(x)=\frac{d}{dx}((\ln x)^3)=3*(\ln x)^2*\frac{d}{dx}(\ln x)=3*(\ln x)^2*\frac{1}{x}=\frac{3(\ln x)^2}{x}.$

2) By evaluating the result of f'(x) at x = h = 3, we get:

$m=f^{\prime}(3)=\frac{3}{3}*(\ln3)^2=(\ln3)^2.$

3) The value of k is:

$k=f(3)=(\ln3)^3$

4) Replacing the values of m, h and k in the general equation of the tangent line, we get:

$y=(\ln3)^2*(x-3)+(\ln3)^3.$

Plotting the function f(x) and the tangent line we verify that our result is correct:

Answer

The equation of the tangent line to f(x) and x = 3 is:

$y=(\ln3)^2*(x-3)+(\ln3)^3$