termine the first derivative of each of the following by using the first principl tukan terbitan pertama bagi setiap yang berikut dengan menggunakan prinsip pertama. Inderstonding of differentiation.

### Example 1: Differentiating \( f(x)=x^2 \) Using First Principles 1. **Write the definition of the derivative:** \[ f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h} \] 2. **Substitute \( f(x)=x^2 \):** \[ f'(x)=\lim_{h\to 0}\frac{(x+h)^2-x^2}{h} \] 3. **Expand \( (x+h)^2 \):** \[ (x+h)^2=x^2+2xh+h^2 \] Thus, \[ f'(x)=\lim_{h\to 0}\frac{x^2+2xh+h^2-x^2}{h} \] 4. **Simplify the expression:** \[ f'(x)=\lim_{h\to 0}\frac{2xh+h^2}{h} = \lim_{h\to 0}\left(2x + h\right) \] 5. **Take the limit as \( h \to 0 \):** \[ f'(x)=2x \] --- ### Example 2: Differentiating \( f(x)=\frac{1}{x} \) Using First Principles 1. **Write the definition of the derivative:** \[ f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h} \] 2. **Substitute \( f(x)=\frac{1}{x} \):** \[ f'(x)=\lim_{h\to 0}\frac{\frac{1}{x+h}-\frac{1}{x}}{h} \] 3. **Combine the fractions in the numerator:** \[ \frac{1}{x+h}-\frac{1}{x}=\frac{x-\left(x+h\right)}{x(x+h)}=\frac{-h}{x(x+h)} \] Thus, the difference quotient becomes \[ f'(x)=\lim_{h\to 0}\frac{\frac{-h}{x(x+h)}}{h} \] 4. **Simplify by canceling \( h \):** \[ f'(x)=\lim_{h\to 0}\frac{-h}{h\,x(x+h)}=\lim_{h\to 0}\frac{-1}{x(x+h)} \] 5. **Take the limit as \( h \to 0 \):** \[ f'(x)=\frac{-1}{x\cdot x}=\frac{-1}{x^2} \] --- These examples show how to compute the derivative using the limit (first principles) definition: - For \( f(x)=x^2 \), we obtained \( f'(x)=2x \). - For \( f(x)=\frac{1}{x} \), we obtained \( f'(x)=-\frac{1}{x^2} \).