Find the derivatives of the function 𝑓(𝑥) = −6𝑥2 − 9𝑥 − 7 by using the concept of limits.

The derivative of a function can be found using the concept of limits. Here's how you can find the derivative of the function f(x) = -6x^2 - 9x - 7.

The derivative of a function f(x) at a specific point x is defined as:

f'(x) = lim(h->0) [f(x+h) - f(x)] / h

Let's apply this definition to the function f(x) = -6x^2 - 9x - 7.

f(x+h) = -6(x+h)^2 - 9(x+h) - 7 = -6(x^2 + 2xh + h^2) - 9x - 9h - 7 = -6x^2 - 12xh - 6h^2 - 9x - 9h - 7

So,

f(x+h) - f(x) = -6x^2 - 12xh - 6h^2 - 9x - 9h - 7 - (-6x^2 - 9x - 7) = -12xh - 6h^2 - 9h

Now, divide this by h:

[f(x+h) - f(x)] / h = -12x - 6h - 9

Now, take the limit as h approaches 0:

f'(x) = lim(h->0) [-12x - 6h - 9] = -12x - 9

So, the derivative of the function f(x) = -6x^2 - 9x - 7 is f'(x) = -12x - 9.