Question

Найти производную функцию f(x)=x (в степени 2) умножить на 5(в степени2x)

Answer 1

F'=2x*5^(2x)+x^2*5^(2x)*ln5*2

Answer 2

Derivative of the Function f(x) = x^2 * 5^(2x)

To find the derivative of the function f(x) = x^2 * 5^(2x), we can use the rules of differentiation. The derivative of a function represents the rate of change of the function with respect to the variable x.

The derivative of the function f(x) = x^2 * 5^(2x) can be calculated using the product rule and the chain rule of differentiation.

Using the product rule, the derivative of the function f(x) = x^2 * 5^(2x) can be expressed as: $$f'(x) = (x^2)' * 5^(2x) + x^2 * (5^(2x))'$$

Now, let's calculate the derivatives of the individual components of the function using the given information.

From Source we have a snippet that provides information on the derivative of a function in a similar format. However, it does not directly provide the derivative of the specific function f(x) = x^2 * 5^(2x). Therefore, we will proceed with the calculation based on the standard rules of differentiation.

The derivative of x^2 with respect to x is 2x, and the derivative of 5^(2x) with respect to x involves the chain rule and logarithmic differentiation.

Using the chain rule, the derivative of 5^(2x) with respect to x can be expressed as: $$\frac{d}{dx} (5^{2x}) = (5^{2x}) * \ln(5) * 2$$

Therefore, the derivative of the function f(x) = x^2 * 5^(2x) is: $$f'(x) = 2x * 5^(2x) + x^2 * 5^(2x) * \ln(5) * 2$$

Simplifying the expression, we get: $$f'(x) = 2x * 5^(2x) + 2x^2 * 5^(2x) * \ln(5)$$

So, the derivative of the function f(x) = x^2 * 5^(2x) with respect to x is 2x * 5^(2x) + 2x^2 * 5^(2x) * ln(5).