Differentiation from First Principles

1. Differentiate the function; y = x
2. Differentiate the function; y = x²
3. Find the derivative of         y =  1/x
4. Find the derivative of         y = sinx

Differentiation from General Formula

5. Differentiate y = 5x³ – 7x² + 100x + 9
6. Find the gradient of the function: y = 3x² + 5x – 7 at x = 1

Product Rule of Differentiation

7. If y = (2x + 5)(6x² + 7x – 1),     find dy/dx
8. If y = x  ,       find dy/dx

Composite Rule of Differentiation

9. If y = (3x + 5)⁷ ,       find dy/dx
10. If y =  ,       find dy/dx

Quotient Rule of Differentiation

11. If y =  ,     find dy/dx

Maximum and Minimum Points

12. For the function y = x² – 5x

Find:

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

 

13. Distinguish between the maximum and minimum points of the function 12x – x³. Find also the maximum and minimum values.

 

14. For the function

Y = x³ – 5x² + 7x + 2

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

Application of Differentiation in Business Studies

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

P = 175 – 4q

And the total cost of the product is

C = 1980 + 2q + 0.02q²

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.

 

16. 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.

 

17. 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.

 

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

Y = 2x³  3 – 15x² + 88x

When would the company have the:

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

19. 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?

 

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

3p = 9q² – 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

21. 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?

 

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

P = 50q – q² – 800

While the total cost of producing the juice is:

C = 800 + 700q – 3q² + q³

• 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.