Question

derive e^2x +3/ e^x

Answer 1

To derive the function e^2x + 3/e^x, we can use the quotient rule.

Let f(x) = e^2x and g(x) = 3/e^x.

Using the quotient rule, the derivative of f(x)/g(x) is given by:

[f'(x) * g(x) - f(x) * g'(x)] / [g(x)]^2

Now, let's find the derivatives of f(x) and g(x):

f'(x) = d/dx (e^2x) = 2e^2x

g'(x) = d/dx (3/e^x) = (0 - 3e^x) / (e^x)^2 = -3e^x / e^2x

Plugging these values back into the quotient rule formula:

[f'(x) * g(x) - f(x) * g'(x)] / [g(x)]^2

= [2e^2x * (3/e^x) - e^2x * (-3e^x / e^2x)] / [(3/e^x)]^2

Simplifying:

= [6e^x - (-3e^x)] / (3^2 / (e^x)^2)

= (6e^x + 3e^x) / (9 / e^2x)

= 9e^x / (9 / e^2x)

= e^2x

Therefore, the derivative of e^2x + 3/e^x is just e^2x.