#### Differentiation from First Principles

- Differentiate the function; y = x
- Differentiate the function; y = x
^{2} - Find the derivative of y = 1/x
- Find the derivative of y = sinx

#### Differentiation from General Formula

- Differentiate y = 5x
^{3}– 7x^{2 }+ 100x + 9 - Find the gradient of the function: y = 3x
^{2}+ 5x – 7 at x = 1

#### Product Rule of Differentiation

- If y = (2x + 5)(6x
^{2}+ 7x – 1), find dy/dx - If y = x , find dy/dx

#### Composite Rule of Differentiation

- If y = (3x + 5)
^{7}, find dy/dx - If y = , find dy/dx

#### Quotient Rule of Differentiation

- If y = , find dy/dx

#### Maximum and Minimum Points

- For the function y = x
^{2 }– 5x

Find:

- The turning point.
- The maximum or minimum point.
- The maximum or minimum value.

- Distinguish between the maximum and minimum points of the function 12x – x
^{3}. Find also the maximum and minimum values.

- For the function

Y = x^{3} – 5x^{2} + 7x + 2

Determine the turning points and distinguish between them. Find also the maximum and minimum values.

#### Application of Differentiation in Business Studies

- The price (p) of a product is given by:

P = 175 – 4q

And the total cost of the product is

C = 1980 + 2q + 0.02q^{2}

Determine:

- The revenue function.
- The revenue when quantity is 6.
- The marginal revenue.
- When the marginal revenue equals zero.
- The marginal cost.
- When the marginal revenue equals marginal cost.

- CharllyCARES Nigeria Ltd. Has expressed the monthly demand function of one of the company’s products as:

P = 3(12 – q)

Where p = unit price (N) and q = quantity demanded while the cost of production C is expressed as:

C = 15(12 – q).

Find:

- The total monthly revenue of the company in terms of the quantity demanded.
- The monthly profit of the company.
- The break-even point(s) and explain the significance as related to the profit of Charlly’s company.

- The price (p) of a commodity in a company is given by

P = 120 – and the total cost of the commodity in a month of the year 2002 is C = 40x where X is the quantity demanded.

Determine:

- The revenue function.
- The revenue when x = 50 units.
- The marginal revenue.
- The profit function.
- The profit when x = 105 units.
- When the profit is maximized.

- Soft Study Nigeria Ltd started a library business in June 2000. If the stock of books, y, after x months of production is:

Y = 2x^{3} 3 – 15x^{2 }+ 88x

When would the company have the:

- The largest number of books and how many?
- Least number of books and how many?

- A company newly bought machine is said to be depreciating. So after x years, its value is expressed as:

Y = 16500e^{– 0.2x}

- At what rate will the machine depreciate after 5 years?
- At what percentage rate will the value depreciate after the five years?

- The demand for the product of a company is given by:

3p = 9q^{2} – 540

While the supply function is given by:

P = 30q – 63

Determine:

- The equilibrium price and quantity.
- The elasticity of supply when q = 5 units

- The demand for one of a company’s product is exponentially distributed with:

Y = 150e^{– 0.02p}

Units per day at a market price of p Naira. What price will there be the greatest customer’s demand?

- The price of Ada’s juice per unit is given by:

P = 50q – q^{2} – 800

While the total cost of producing the juice is:

C = 800 + 700q – 3q^{2} + q^{3}

- Determine the quantity which maximizes the profit.
- How many units of the juice needs to be produced in order to maximize profit if the selling price remains at N700.