📚 Part 1: Understanding Derivatives

1. What does the derivative of a function represent?

The area under the curve

The instantaneous rate of change

The maximum value of the function

The y-intercept of the function

2. Which notation represents the derivative of f(x)?

f'(x)

dy/dx

df/dx

∫f(x)dx

3. The derivative from first principles is defined as:

f(x + h) - f(x)

lim[h→0] (f(x+h) - f(x))/h

f(x)/h

h × f(x)

4. Using the power rule, what is the derivative of f(x) = x³?

3x

x²

3x²

x³/3

✏️ Part 2: Calculating Derivatives

5. Find the derivative of each function using the power rule:

a) f(x) = x⁵ f'(x) = ___________

b) g(x) = 2x⁴ g'(x) = ___________

c) h(x) = x² h'(x) = ___________

d) k(x) = 7x k'(x) = ___________

6. A particle's position is given by s(t) = t² + 3t metres, where t is time in seconds. Find the velocity function v(t).

7. If f(x) = x³ - 2x² + 5x, find f'(x) and evaluate f'(2).

🌟 Part 3: Real-World Applications

8. A bacteria culture grows according to P(t) = 100e^(0.5t), where P is the population and t is time in hours. What does P'(t) represent in this context?

9. A car travels along a straight road. Its position function is s(t) = 2t³ - 12t² + 18t metres after t seconds.

a) Find the velocity function: v(t) = ___________

b) What is the velocity at t = 3 seconds? ___________

c) At what time(s) is the car at rest? ___________

10. Explain in your own words what it means when the derivative of a function equals zero at a particular point.

Introduction to Differentiation

Introduction to Differentiation

📚 Part 1: Understanding Derivatives

✏️ Part 2: Calculating Derivatives

🌟 Part 3: Real-World Applications

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