Question

Find an equation of the tangent line, at the designated point, to each of the following: f(x)=xlnx,x=e.

Answer 1

The equation of the tangent line to the function f(x)=xlnx at x=e is y - (e)(ln(e)) = 2(x - e).

Step-by-step explanation:

First, we find the derivative of the function f(x)=xlnx using the product rule. The derivative is f'(x) = (1)(lnx) + (x)(1/x) = lnx + 1.

Next, we find the slope of the tangent line at x=e by evaluating f'(e) = ln(e) + 1 = 1 + 1 = 2.

Finally, we use the point-slope form of the equation of a line, y - y1 = m(x - x1), where (x1, y1) is the point of tangency, x=e, and the slope, m, is 2. Substituting the values, we get the equation of the tangent line as y - (e)(ln(e)) = 2(x - e).