step 1\nif two numbers have a difference of 198, and one of them is ( x + 198 ), then the other is…

step 1\nif two numbers have a difference of 198, and one of them is ( x + 198 ), then the other is +396\n\nstep 2\nthe product of two numbers ( x ) and ( x + 198 ) can be simplified to be ( x^{2}+198x ).\n\nstep 3\nif ( f(x)=x^{2}+198x ), then ( f(x)=)

step 1\nif two numbers have a difference of 198, and one of them is ( x + 198 ), then the other is +396\n\nstep 2\nthe product of two numbers ( x ) and ( x + 198 ) can be simplified to be ( x^{2}+198x ).\n\nstep 3\nif ( f(x)=x^{2}+198x ), then ( f(x)=)

Answer

Explanation:

Step1: Apply the power rule of differentiation

The power rule states that if (y = x^n), then (y^\prime=nx^{n - 1}). For the function (f(x)=x^{2}+198x), differentiate each term separately. For the term (x^{2}), using the power rule with (n = 2), we have (\frac{d}{dx}(x^{2})=2x^{2-1}=2x).

Step2: Apply the constant - multiple rule of differentiation

The constant - multiple rule states that if (y = ax) (where (a) is a constant), then (y^\prime=a). For the term (198x), using the constant - multiple rule with (a = 198), we have (\frac{d}{dx}(198x)=198\times\frac{d}{dx}(x)). Since (\frac{d}{dx}(x)=1), this term differentiates to (198).

Step3: Sum the derivatives of the terms

By the sum rule of differentiation (if (y = u + v), then (y^\prime=u^\prime + v^\prime)), for (u=x^{2}) and (v = 198x), (f^\prime(x)=\frac{d}{dx}(x^{2})+\frac{d}{dx}(198x)). Substituting the derivatives from Step1 and Step2, we get (f^\prime(x)=2x + 198).

Answer:

(2x+198)