Calculus question?

Question

Find the derivative for the function f(x)=e^(2x)/X^2

Answer 1

f = u / v
f' = (u'v - uv') / v^2

u = e^2x, u' = 2e^2x
v = x^2, v' = 2x

so

f' = (2x^2 * e^2x - 2x * e^2x) / x^4 =
2x * e^2x (x - 1) / x^4 =
2 * e^2x (x - 1) / x^3

Answer 2

Use the product rule on this.

That is split the entity into 2 pieces, call them u and v. The product rule says that the derivative of the expression is

u'v+v'u

So, let u = e^2x and let v=x^-2

Now, du/dx = 2e^2x and dv/dx = -2x^-3

So, put it all together:

(2e^2x * x^-2) - (2e^2x * x^-3)

Simplified, this is (2e^2x/x^2) - (2e^2x/x^3)

The lowest common denominator is x^3, so multiply the left term by x/x, and you get:

(2xe^2x-2e^2x)/x^3 or

(2e^2x * (x-1))/x^3

Answer 3

The derivative is (2x^2e^(2x)-2xe^(2x))/x^4.
Simplify and get 2(xe^(2x)-e^(2x))/x^3.

Answer 4

f´(x) =2e^2x*1/x^2-2*e^2x*1/x^3=2e^2x(1/x^2-1/x^3)