Find the Function g'(x)=3x^2-2x-4

Problem

g(x)^′=3*x^2−2*x−4

Solution

1. Identify the task as finding the general antiderivative (the function g(x) of the given derivative g(x)^′

2. Apply the power rule for integration, which states that (∫_^)(x^n*d(x))=(x^(n+1))/(n+1) for n≠−1

3. Integrate each term of the expression 3*x^2−2*x−4 with respect to x

4. Calculate the integral of the first term: (∫_^)(3*x^2*d(x))=3⋅(x^3)/3=x^3

5. Calculate the integral of the second term: (∫_^)(−2*x*d(x))=−2⋅(x^2)/2=−x^2

6. Calculate the integral of the constant term: (∫_^)(−4*d(x))=−4*x

7. Add the constant of integration C to represent the family of functions that share this derivative.

Final Answer

g(x)=x^3−x^2−4*x+C

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