Question

if ( f(x)=3 + 3x-5x^{2} ), find ( f^{prime}(3) ).

Explanation:

Step1: Differentiate the function

Differentiate \(f(x) = 3+3x - 5x^{2}\) term - by - term.
Using the power rule \(\frac{d}{dx}(x^{n})=nx^{n - 1}\) and \(\frac{d}{dx}(c)=0\) (where \(c\) is a constant).
\(f^{\prime}(x)=\frac{d}{dx}(3)+\frac{d}{dx}(3x)-\frac{d}{dx}(5x^{2})\)
\(f^{\prime}(x)=0 + 3-5\times2x\)
\(f^{\prime}(x)=3 - 10x\)

Step2: Evaluate the derivative at \(x = 3\)

Substitute \(x = 3\) into \(f^{\prime}(x)\).
\(f^{\prime}(3)=3-10\times3\)
\(f^{\prime}(3)=3 - 30\)

Answer:

\(- 27\)