Question

Решить производную ((2х-1)/(е^2х))

Answer 1

Решение задания смотри на фотографии

Answer 2

To find the derivative of the given function, you can use the quotient rule, which states that the derivative of a quotient of two functions is given by:

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

where f(x) is the numerator function and g(x) is the denominator function.

Let's apply the quotient rule to find the derivative of ((2x - 1) / (e^(2x))):

f(x) = 2x - 1 g(x) = e^(2x)

Now, we need to find the derivatives of f(x) and g(x):

f'(x) = d/dx (2x - 1) = 2

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

Now, plug these values into the quotient rule formula:

d/dx [((2x - 1) / (e^(2x)))] = [e^(2x) * 2 - (2x - 1) * 2e^(2x)] / [e^(2x)]^2

Simplify the expression:

d/dx [((2x - 1) / (e^(2x)))] = (2e^(2x) - 2(2x - 1)e^(2x)) / (e^(4x))

Now, factor out e^(2x) from the numerator:

d/dx [((2x - 1) / (e^(2x)))] = e^(2x) * (2 - 2(2x - 1)) / (e^(4x))

Finally, simplify the expression further:

d/dx [((2x - 1) / (e^(2x)))] = (2 - 4x + 2) / (e^(4x))

d/dx [((2x - 1) / (e^(2x)))] = (-4x + 4) / (e^(4x))

So, the derivative of ((2x - 1) / (e^(2x))) is (-4x + 4) / (e^(4x)).