Question

Consider the function f(x)=2ˣ³+6ˣ²−18x−54 a. Find the first derivative. fγ ⁶.

Answer 1

To find the first derivative of the function f(x)=2ˣ³+6ˣ²−18x−54, apply the power rule to each term. The first derivative, or f'(x), is 6x² + 12x - 18.

Step-by-step explanation:

The subject of the question is finding the first derivative of a given function. Since the function provided is f(x)=2ˣ³+6ˣ²−18x−54, to find the first derivative, we would apply the power rule for differentiation. The derivative, often symbolized with f'(x) or ƒγ, represents the rate at which the function's value is changing at any given point x.

To find the first derivative of f(x), we differentiate each term separately:

1. For 2ˣ³, the derivative is 6x² (as per the power rule, n*x^(n-1)).
2. For 6ˣ², the derivative is 12x.
3. For −18x, the derivative is −18.
4. For the constant −54, the derivative is 0, since the derivative of a constant is zero.

Combining these results, the first derivative of the function f(x) is f'(x) = 6x² + 12x - 18.