Question

for the function $f(x)=2x^3 - 3x^2 + 5x + 3$, find $f(x)$. then find $f$
to find $f(x)$, first find $f(x)$.
$f(x)= 6x^2 - 6x + 5$
find $f(x)$.
$f(x)=\square$

Explanation:

Step1: Find \( f''(x) \) from \( f'(x) \)

Given \( f'(x) = 6x^2 - 6x + 5 \), we differentiate term - by - term. The power rule for differentiation is \( \frac{d}{dx}(x^n)=nx^{n - 1} \) and the derivative of a constant is 0.
For the term \( 6x^2 \), using the power rule, the derivative is \( 6\times2x^{2 - 1}=12x \).
For the term \( - 6x \), using the power rule, the derivative is \( - 6\times1x^{1 - 1}=-6 \).
For the constant term \( 5 \), the derivative is 0.
So, \( f''(x)=12x - 6 \).

Step2: Find \( f'''(x) \) from \( f''(x) \)

Now, we differentiate \( f''(x)=12x - 6 \) term - by - term.
For the term \( 12x \), using the power rule, the derivative is \( 12\times1x^{1 - 1}=12 \).
For the constant term \( - 6 \), the derivative is 0.
So, \( f'''(x) = 12 \).

Answer:

\( 12 \)