Question

How do you find the equation of the tangent line to the graph of `f(x) = x ln x` at x = 1?

Answer 1

`y=x-1`

Explanation:

To find the equation of the tangent line, we must first find the derivative of the function and then evaluate at the given point, this will give you the slope `m`. Using the chain rule:

`d/dx xlnx = lnxd/dxx+xd/dxlnx`

`\qquad\qquad\qquad\qquad= lnx(1) + x(1/x)`

`\qquad\qquad\qquad\qquad= lnx + 1`

So, at `x=1`, `f(x) = 0` and `f'(x) = m = 1`

This tells us there's a translation of one unit in the x direction, let's prove it:

`y = mx + b`

`0 = (1)(1) + b`

`b = -1`

Finally the equation `y` of the tangent line will be:

`y=x-1`