(e^4x)/(4x^2 -2)find the derivative

Answer 1

To find the derivative of the function (e^4x)/(4x^2 - 2), we can use the quotient rule.

The quotient rule states that if we have a function f(x) = g(x)/h(x), then its derivative is given by:

f'(x) = (g'(x) * h(x) - g(x) * h'(x))/(h(x))^2

Let's apply this rule to our function:

g(x) = e^4x
g'(x) = 4e^4x

h(x) = 4x^2 - 2
h'(x) = 8x

Now, we can substitute these values into the quotient rule formula to find the derivative:

f'(x) = (4e^4x * (4x^2 - 2) - e^4x * 8x)/((4x^2 - 2)^2)

Simplifying this expression, we get:

f'(x) = (4e^4x * (4x^2 - 2) - 8x * e^4x)/((4x^2 - 2)^2)

So, the derivative of the function (e^4x)/(4x^2 - 2) is:
f'(x) = (4e^4x * (4x^2 - 2) - 8x * e^4x)/((4x^2 - 2)^2)