Question

Y=1/x√x-4 Вычислите производную данной функции

Answer 1

решение задания смотри на фотографии

Answer 2

To find the derivative of the given function Y = 1/(x√(x-4)), we can use the rules of differentiation. Let's proceed step by step:

Step 1: Rewrite the function in a more convenient form: Y = x^(-1/2) * (x - 4)^(-1)

Step 2: Apply the product rule of differentiation: If we have a function of the form f(x) = g(x) * h(x), then the derivative of f(x) with respect to x is given by f'(x) = g'(x) * h(x) + g(x) * h'(x).

In our case, g(x) = x^(-1/2) and h(x) = (x - 4)^(-1).

Step 3: Calculate the derivatives of g(x) and h(x):

• Derivative of g(x): g'(x) = d/dx (x^(-1/2)) = (-1/2) * x^((-1/2) - 1) = (-1/2) * x^(-3/2) = -1/(2x^(3/2))

• Derivative of h(x): h'(x) = d/dx ((x - 4)^(-1)) = -1 * (x - 4)^(-1 - 1) = -1/(x - 4)^2

Step 4: Now, apply the product rule and combine the derivatives: Y' = g'(x) * h(x) + g(x) * h'(x) = (-1/(2x^(3/2))) * (x - 4)^(-1) + (x^(-1/2)) * (-1/(x - 4)^2) = -1/(2x^(3/2)(x - 4)) - x^(-1/2)/(x - 4)^2

Thus, the derivative of the function Y = 1/(x√(x-4)) with respect to x is Y' = -1/(2x^(3/2)(x - 4)) - x^(-1/2)/(x - 4)^2.