if f(x)= e^2x/2x, then what is the slope of the line tangent to the graph of f at x=1

Answer 1

To find the slope of the tangent line to the graph of f at x=1, we first need to find the derivative of f(x).

Given f(x) = e^(2x)/2x, we can rewrite it as:

f(x) = (1/2) * e^(2x) * x^(-1)

Now, we can differentiate f(x) using the product rule:

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

Now, we can find the slope of the tangent line at x=1 by plugging in x=1 into f'(x):

f'(1) = -e^2/(2*1^2) + e^2/1
f'(1) = -e^2/2 + e^2
f'(1) = e^2/2

Therefore, the slope of the tangent line to the graph of f at x=1 is e^2/2.