Question

Finding an Equation of a Tangent Line In Exercise, find an equation of the tangent line to the graph of the function at the given point. See Example 5.

y = (ln x)/x; (e, 1/e)

Answer 1

y = 1/e

y = 0.37

Explanation:

y = (ln x)/x; (e, 1/e)

Step 1

Find the point of tangency.

It's given as (e,1/e) or (2.72,0.37)

Step 2

Find the first derivative, and evaluate it at x=e

Using product rule.

d(uv) = udv + vdu

Where u = lnx and v = 1/x

du = 1/x , dv = -1/x^2

f'(x) = -lnx/x^2 +1/x^2

f'(x) = (1-lnx)/x^2

At x = e

f'(e) = (1-lne)/e^2

f'(e) = 0

The slope of the tangent line at this point is m= 0

Step 3

Find the equation of the tangent line at (e,1/e) with a slope of m=0

y−y1 = m(x - x1)

y-(1/e) = 0(x-1)

y = 1/e

y = 0.37