Determine the slope of the tangent at the indicated value of x. State as an exact value.
f(x)= e^x^2 / 2x, x=1
f'(x) = 1/2 (2 - 1/x^2) e^x^2
So plug in x=1
Looks like 1/2 e^2
using the quotient rule:
f '(x) = (2x(2x)e^(x^2) - 2 e^(x^2) )/(4x^2)
= ( e^(x^2) (4x^2 - 2x) )/(4x^2)
when x = 1
f '(1) = e^1 (2)/4
= (1/2)e or e/2
duh - saved again.
To determine the slope of the tangent at a specific value of x for a given function, we need to find the derivative of the function and evaluate it at the given value of x.
Given:
f(x) = e^(x^2) / (2x)
x = 1
Step 1: Find the derivative of the function f(x):
To find the derivative, we can use the quotient rule.
The quotient rule states that for any function f(x) = g(x) / h(x), the derivative f'(x) is given by:
f'(x) = (g'(x) * h(x) - g(x) * h'(x)) / (h(x))^2
For our function f(x) = e^(x^2) / (2x), we have:
g(x) = e^(x^2)
h(x) = 2x
Let's find the derivatives of g(x) and h(x):
g'(x) = d/dx(e^(x^2))
h'(x) = d/dx(2x)
To calculate g'(x), we need to apply the chain rule, which states that if we have a composite function g(h(x)), its derivative is g'(h(x)) * h'(x).
Apply the chain rule to g(x) = e^(x^2):
g'(x) = d/dx(e^(x^2)) = e^(x^2) * d/dx(x^2)
To calculate d/dx(x^2), we can apply the power rule, which states that d/dx(x^n) = n * x^(n-1):
d/dx(x^2) = 2x
Substituting these results back into the quotient rule, we have:
f'(x) = [(e^(x^2) * 2x) - (e^(x^2) * 2)] / (2x)^2
Step 2: Evaluate the derivative at x = 1:
To find the slope of the tangent at x = 1, we substitute x = 1 into the derivative we just found:
f'(1) = [(e^(1^2) * 2 * 1) - (e^(1^2) * 2)] / (2 * 1)^2
f'(1) = (2e - 2e) / 4
f'(1) = 0 / 4
f'(1) = 0
Therefore, the slope of the tangent at x = 1 for the function f(x) = e^(x^2) / (2x) is 0.