Question

Найдите производную функции: у=3х в пятой степени+х в квадрате*корень из х

Answer 1

У'=15х^4 + (х^2* х^1/2)'=15х^4+2х*х^1/2-х^2*1/2х^-1/2

Answer 2

Finding the Derivative of the Function

To find the derivative of the function у = 3х^5 * х^2 * √х, we can use the power rule and the product rule of differentiation.

The power rule states that if we have a function of the form f(x) = x^n, then its derivative is given by f'(x) = n * x^(n-1).

The product rule states that if we have two functions u(x) and v(x), then the derivative of their product is given by (u * v)' = u' * v + u * v'.

Let's apply these rules to find the derivative of the given function:

1. First, let's simplify the function by combining the powers of x: у = 3 * x^(5+2) * √x. - Simplifying further, we have: у = 3 * x^7 * √x.

2. Now, let's find the derivative of each term separately: - The derivative of 3 is 0. - The derivative of x^7 using the power rule is 7 * x^(7-1) = 7 * x^6. - The derivative of √x is (1/2) * x^(-1/2) using the power rule.

3. Applying the product rule, we have: - The derivative of у = 3 * x^7 * √x is у' = 0 * √x + 3 * 7 * x^6 * (1/2) * x^(-1/2). - Simplifying further, we get: у' = 0 + (21/2) * x^6 * x^(-1/2). - Combining the terms, we have: у' = (21/2) * x^(6 - 1/2). - Simplifying the exponent, we get: у' = (21/2) * x^(11/2).

Therefore, the derivative of the function у = 3х^5 * х^2 * √х is у' = (21/2) * x^(11/2).

Please note that the above derivation assumes that the function is defined for all real values of x. If there are any specific restrictions on the domain of the function, please let me know.

Let me know if there's anything else I can help you with!