How do you find the equation of the line tangent to the graph of #f(x)= (ln x)^5# at x=5?

Question

Paisley Johnson · Accepted Answer

#f ' (x) = 5 (ln x ) (1/x)#
#f ' (5) = 5 (ln 5) (1/5) = ln 5# ---- this is the slope
#f(5) = (ln 5 )^5#

#y-(ln 5)^5 = ln 5 (x - 5)# Use chain rule to find derivative of f(x) and then put in 5 for x. Find the y-coordinate by putting in 5 for x in the original function then use the slope and the point to write the equation of a tangent line.

Aurora Alvarado · Answer

undefined To find the equation of the line tangent to the graph of f(x) = (ln x)^5 at x = 5, we can follow these steps:

1. Find the derivative of f(x) using the chain rule. The derivative of (ln x)^5 is 5(ln x)^4 * (1/x).

2. Evaluate the derivative at x = 5 to find the slope of the tangent line. Substitute x = 5 into the derivative expression and calculate the result.

3. Use the point-slope form of a line to write the equation of the tangent line. The point-slope form is y - y1 = m(x - x1), where (x1, y1) is the point of tangency and m is the slope.

4. Substitute the values of x1, y1, and m into the point-slope form to obtain the equation of the tangent line.

By following these steps, you can find the equation of the line tangent to the graph of f(x) = (ln x)^5 at x = 5.