To find the instantaneous rate of change of the function [tex]\( f(x) = -2x^2 - 5x \)[/tex] at [tex]\( x = 3 \)[/tex], we need to follow these steps:
### Step 1: Find the derivative of [tex]\( f(x) \)[/tex]
The instantaneous rate of change of the function at a particular point is given by the derivative of the function at that point. So, we first need to find the derivative of [tex]\( f(x) \)[/tex].
The function is:
[tex]\[ f(x) = -2x^2 - 5x \][/tex]
To find the derivative, [tex]\( f'(x) \)[/tex], we differentiate [tex]\( f(x) \)[/tex] with respect to [tex]\( x \)[/tex].
The differentiation steps are:
1. Differentiate [tex]\( -2x^2 \)[/tex]:
[tex]\[
\frac{d}{dx} (-2x^2) = -2 \cdot 2x = -4x
\][/tex]
2. Differentiate [tex]\( -5x \)[/tex]:
[tex]\[
\frac{d}{dx} (-5x) = -5
\][/tex]
Therefore, the derivative of the function [tex]\( f(x) \)[/tex] is:
[tex]\[ f'(x) = -4x - 5 \][/tex]
### Step 2: Evaluate the derivative at [tex]\( x = 3 \)[/tex]
Now we need to find the value of the derivative at [tex]\( x = 3 \)[/tex].
Substitute [tex]\( x = 3 \)[/tex] into the derivative [tex]\( f'(x) \)[/tex]:
[tex]\[
f'(3) = -4(3) - 5
\][/tex]
Calculate the value:
[tex]\[
f'(3) = -12 - 5 = -17
\][/tex]
### Conclusion
The instantaneous rate of change of the function [tex]\( f(x) = -2x^2 - 5x \)[/tex] at [tex]\( x = 3 \)[/tex] is [tex]\( -17 \)[/tex].
