SULIT 15. (a) (i) Cari $$ f^{\prime}(x) $$ bagi $$ f(x)=3 x^{2}+1 $$ dengan menggunakan pembezaan dengan prinsip pertama. Find $$ f^{\prime}(x) $$ for $$ f(x)=3 x^{2}+1 $$ using differentiation by the first principles. (ii) Cari terbitan pertama bagi fungsi $$ f(x)=\frac{1}{2} x^{2}-3 x $$. Find the first derivative of the function $$ f(x)=\frac{1}{2} x^{2}-3 x $$.

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**(a)(i) Finding $$ f^{\prime}(x) $$ for $$ f(x) = 3x^2 + 1 $$ using first principles**

We use the limit definition of the derivative:
$$
f^{\prime}(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
$$

1. Compute $$ f(x+h) $$:
   $$
   f(x+h) = 3(x+h)^2 + 1 = 3(x^2 + 2xh + h^2) + 1 = 3x^2 + 6xh + 3h^2 + 1
   $$

2. Compute the difference $$ f(x+h) - f(x) $$:
   $$
   f(x+h) - f(x) = (3x^2 + 6xh + 3h^2 + 1) - (3x^2 + 1) = 6xh + 3h^2
   $$

3. Divide by $$ h $$:
   $$
   \frac{f(x+h) - f(x)}{h} = \frac{6xh + 3h^2}{h} = 6x + 3h
   $$

4. Take the limit as $$ h \to 0 $$:
   $$
   f^{\prime}(x) = \lim_{h \to 0}(6x + 3h) = 6x
   $$

Thus, 
$$
f^{\prime}(x) = 6x.
$$

---

**(a)(ii) Finding the first derivative of $$ f(x) = \frac{1}{2}x^2 - 3x $$**

We differentiate term by term using standard differentiation rules.

1. The derivative of $$ \frac{1}{2}x^2 $$ is:
   $$
   \frac{d}{dx}\left(\frac{1}{2}x^2\right) = \frac{1}{2} \cdot 2x = x.
   $$

2. The derivative of $$ -3x $$ is:
   $$
   \frac{d}{dx}(-3x) = -3.
   $$

Combine the results:
$$
f^{\prime}(x) = x - 3.
$$
**(a)(i)** $$ f^{\prime}(x) = 6x $$ **(a)(ii)** $$ f^{\prime}(x) = x - 3 $$

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