Consider the function f(x)=In(x^2+1). Calculate the slope of the tangent line at the point
(2,ln(5)). Round your answer to two decimal places.
slope = rise/run
tan : opposite/hypotneus
To calculate the slope of the tangent line at a given point, we need to find the derivative of the function at that point. In this case, we have the function f(x) = ln(x^2 + 1) and want to find the slope at the point (2, ln(5)).
Step 1: Find the derivative of the function f(x).
The derivative of ln(x) is 1/x, and the chain rule applies when we have a composition of functions like ln(x^2 + 1). Applying the chain rule, we get:
f'(x) = 1/(x^2 + 1) * (2x) = 2x/(x^2 + 1)
Step 2: Substitute x = 2 into the derivative to find the slope at the given point.
Plugging x = 2 into f'(x), we get:
f'(2) = 2(2)/(2^2 + 1) = 4/5 = 0.80
Rounding to two decimal places, the slope of the tangent line at the point (2, ln(5)) is approximately 0.80.