Question

Найдите производную функции y=4-15x⁶ и вычислите ее значение в точке x=?

Answer 1

У⁾(х) = - 90х⁵
у⁾(5)= -90·5⁵=-281250

Answer 2

Calculating the Derivative of the Function and Its Value at a Specific Point

To find the derivative of the function \( y = 4 - 15x^6 \) and compute its value at a specific point \( x = a \), we can follow these steps:

1. Find the Derivative of the Function: The derivative of a function represents the rate of change of the function with respect to its variable. In this case, we need to find the derivative of \( y = 4 - 15x^6 \) with respect to \( x \).

The derivative of a constant is 0, and the derivative of \( x^n \) is \( nx^{n-1} \). Therefore, the derivative of the given function is: \[ \frac{dy}{dx} = 0 - 15 \times 6x^{6-1} = -90x^5 \]

This means that the derivative of the function \( y = 4 - 15x^6 \) with respect to \( x \) is \( -90x^5 \) [[1]].

2. Compute the Value of the Derivative at the Given Point: To find the value of the derivative at a specific point \( x = a \), we simply substitute \( a \) into the derivative function \( -90x^5 \).

Therefore, the value of the derivative at the point \( x = a \) is: \[ \frac{dy}{dx} \bigg|_{x=a} = -90a^5 \]

This gives us the value of the derivative of the function \( y = 4 - 15x^6 \) at the point \( x = a \) as \( -90a^5 \).

So, in summary, the derivative of the function \( y = 4 - 15x^6 \) with respect to \( x \) is \( -90x^5 \), and the value of the derivative at the point \( x = a \) is \( -90a^5 \).