# Derive By First Principle – Example 1

## Solving Equation of Index Number

#### Example

Derive the equationÂ $$y = {x^2} + \frac{3}{x}$$ by using first principle.

\eqalign{ & y = {x^2} + \frac{3}{x} \cr & \frac{{dy}}{{dx}} = \mathop {\lim }\limits_{\delta x \to 0} \frac{{\delta y}}{{\delta x}} \cr & = \mathop {\lim }\limits_{\delta x \to 0} \frac{{\left( {y + \delta y} \right) – y}}{{\delta x}} \cr & = \mathop {\lim }\limits_{\delta x \to 0} \frac{{\left[ {{{\left( {x + \delta x} \right)}^2} + \frac{3}{{x + \delta x}}} \right] – \left[ {{x^2} + \frac{3}{x}} \right]}}{{\delta x}} \cr & = \mathop {\lim }\limits_{\delta x \to 0} \frac{{{x^2} + 2x\delta x + \delta {x^2} – {x^2} + \frac{3}{{x + \delta x}} – \frac{3}{x}}}{{\delta x}} \cr & = \mathop {\lim }\limits_{\delta x \to 0} \frac{{2x\delta x + \delta {x^2}}}{{\delta x}} + \frac{{\frac{{3x – 3(x + \delta x)}}{{x(x + \delta x)}}}}{{\delta x}} \cr & = \mathop {\lim }\limits_{\delta x \to 0} 2x + \delta x + \frac{{ -3 \delta x}}{{x(x + \delta x)}} \times \frac{1}{{\delta x}} \cr & = \mathop {\lim }\limits_{\delta x \to 0} 2x + \delta x + \frac{{ – 3}}{{x(x + \delta x)}} \cr & = 2x – \frac{3}{{{x^2}}} \cr}

$$\frac{{dy}}{{dx}} = \mathop {\lim }\limits_{\delta x \to 0} \frac{{\delta y}}{{\delta x}}$$

# Selesaikan Persamaan Nombor Indeks – Contoh 1

## Selesaikan Persamaan Nombor Indeks

#### Contoh

Selesaikan persamaan 5x-1 + 5x+2 = 3150

\eqalign{ & {5^{x – 1}} + {5^{x + 2}} = 3150 \cr & \frac{{{5^x}}}{5} + {5^x} \times {5^2} = 3150 \cr & \frac{{{5^x}}}{5} + \frac{{125 \times {5^x}}}{5} = 3150 \cr & 126 \times {5^x} = 5 \times 3150 \cr & {5^x} = \frac{{5 \times 3150}}{{126}} \cr & {5^x} = 125 = {5^3} \cr & x = 3 \cr}

\eqalign{ & {a^m} \times {a^n} = {a^{m + n}} \cr & {a^m} \div {a^n} = \frac{{{a^m}}}{{{a^n}}} = {a^{m – n}} \cr & {\text{Hence}} \cr & {5^{x – 1}} = {5^x} \div 5 = \frac{{{5^x}}}{5} \cr & {5^{x + 2}} = {5^x} \times {5^2} \cr}

# Solving Equation of Index Number – Example 1

## Solving Equation of Index Number

#### Example

Solve the equation 5x-1 + 5x+2 = 3150

\eqalign{ & {5^{x – 1}} + {5^{x + 2}} = 3150 \cr & \frac{{{5^x}}}{5} + {5^x} \times {5^2} = 3150 \cr & \frac{{{5^x}}}{5} + \frac{{125 \times {5^x}}}{5} = 3150 \cr & 126 \times {5^x} = 5 \times 3150 \cr & {5^x} = \frac{{5 \times 3150}}{{126}} \cr & {5^x} = 125 = {5^3} \cr & x = 3 \cr}

\eqalign{ & {a^m} \times {a^n} = {a^{m + n}} \cr & {a^m} \div {a^n} = \frac{{{a^m}}}{{{a^n}}} = {a^{m – n}} \cr & {\text{Hence}} \cr & {5^{x – 1}} = {5^x} \div 5 = \frac{{{5^x}}}{5} \cr & {5^{x + 2}} = {5^x} \times {5^2} \cr}